# 37 Group Libraries

When you start GAP it already knows several groups. Currently GAP initially knows the following groups: item some basic groups, such as cyclic groups or symmetric groups (see The Basic Groups Library), The Primitive Groups Library), The Transitive Groups Library), The Solvable Groups Library), item the 2-groups of size at most 256 (see The 2-Groups Library), item the 3-groups of size at most 729 (see The 3-Groups Library), item the irreducible solvable subgroups of GL(n,p) for n > 1 and The Irreducible Solvable Linear Groups Library), item the finite perfect groups of size at most 10^6 (excluding 11 sizes) (see The Library of Finite Perfect Groups), item the irreducible maximal finite integral matrix groups of dimension Irreducible Maximal Finite Integral Matrix Groups), The Crystallographic Groups Library). item the groups of order at most 1000 except for 512 and 768 (see The Small Groups Library).

Each of the set of groups above is called a library. The whole set of groups that GAP knows initially is called the GAP collection of group libraries. There is usually no relation between the groups in the different libraries.

Several of the libraries are accessed in a uniform manner. For each of these libraries there is a so called selection function that allows you to select the list of groups that satisfy given criterias from a library. The example function allows you to select one group that satisfies given criteria from the library. The low-level extraction function allows you to extract a single group from a library, using a simple Selection Functions, Example Functions, and Extraction Functions.

Note that a system administrator may choose to install all, or only a few, or even none of the libraries. So some of the libraries mentioned below may not be available on your installation.

## 37.1 The Basic Groups Library

`CyclicGroup( n )` `CyclicGroup( D, n )`

In the first form `CyclicGroup` returns the cyclic group of size n as a permutation group. In the second form D must be a domain of group elements, e.g., `Permutations` or `AgWords`, and `CyclicGroup` returns the cyclic group of size n as a group of elements of that type.

```    gap> c12 := CyclicGroup( 12 );
Group( ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12) )
gap> c105 := CyclicGroup( AgWords, 5*3*7 );
Group( c105_1, c105_2, c105_3 )
gap> Order(c105,c105.1); Order(c105,c105.2); Order(c105,c105.3);
105
35
7 ```

`AbelianGroup( sizes )` `AbelianGroup( D, sizes )`

In the first form `AbelianGroup` returns the abelian group C_{sizes[1]} * C_{sizes[2]} * ... * C_{sizes[n]}, where sizes must be a list of positive integers, as a permutation group. In the second form D must be a domain of group elements, e.g., `Permutations` or `AgWords`, and `AbelianGroup` returns the abelian group as a group of elements of this type.

```    gap> g := AbelianGroup( AgWords, [ 2, 3, 7 ] );
Group( a, b, c )
gap> Size( g );
42
gap> IsAbelian( g );
true ```

The default function `GroupElementsOps.AbelianGroup` uses the functions `CyclicGroup` and `DirectProduct` (see DirectProduct) to construct the abelian group.

`ElementaryAbelianGroup( n )` `ElementaryAbelianGroup( D, n )`

In the first form `ElementaryAbelianGroup` returns the elementary abelian group of size n as a permutation group. n must be a positive prime power of course. In the second form D must be a domain of group elements, e.g., `Permutations` or `AgWords`, and `ElementaryAbelianGroup` returns the elementary abelian group as a group of elements of this type.

```    gap> ElementaryAbelianGroup( 16 );
Group( (1,2), (3,4), (5,6), (7,8) )
gap> ElementaryAbelianGroup( AgWords, 3 ^ 10 );
Group( m59049_1, m59049_2, m59049_3, m59049_4, m59049_5, m59049_6,
m59049_7, m59049_8, m59049_9, m59049_10 ) ```

The default function `GroupElementsOps.ElementaryAbelianGroup` uses `CyclicGroup` and `DirectProduct` (see DirectProduct to construct the elementary abelian group.

`DihedralGroup( n )` `DihedralGroup( D, n )`

In the first form `DihedralGroup` returns the dihedral group of size n as a permutation group. n must be a positive even integer. In the second form D must be a domain of group elements, e.g., `Permutations` or `AgWords`, and `DihedralGroup` returns the dihedral group as a group of elements of this type.

```    gap> DihedralGroup( 12 );
Group( (1,2,3,4,5,6), (2,6)(3,5) ) ```

`PolyhedralGroup( p, q )` `PolyhedralGroup( D, p, q )`

In the first form `PolyhedralGroup` returns the polyhedral group of size `p * q` as a permutation group. p and q must be positive integers and there must exist a nontrivial p-th root of unity modulo every prime factor of q. In the second form D must be a domain of group elements, e.g., `Permutations` or `Words`, and `PolyhedralGroup` returns the polyhedral group as a group of elements of this type.

```    gap> PolyhedralGroup( 3, 13 );
Group( ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13), ( 2, 4,10)( 3, 7, 6)
( 5,13,11)( 8, 9,12) )
gap> Size( last );
39 ```

`SymmetricGroup( d )` `SymmetricGroup( D, d )`

In the first form `SymmetricGroup` returns the symmetric group of degree d as a permutation group. d must be a positive integer. In the second form D must be a domain of group elements, e.g., `Permutations` or `Words`, and `SymmetricGroup` returns the symmetric group as a group of elements of this type.

```    gap> SymmetricGroup( 8 );
Group( (1,8), (2,8), (3,8), (4,8), (5,8), (6,8), (7,8) )
gap> Size( last );
40320 ```

`AlternatingGroup( d )` `AlternatingGroup( D, d )`

In the first form `AlternatingGroup` returns the alternating group of degree d as a permutation group. d must be a positive integer. In the second form D must be a domain of group elements, e.g., `Permutations` or `Words`, and `AlternatingGroup` returns the alternating group as a group of elements of this type.

```    gap> AlternatingGroup( 8 );
Group( (1,2,8), (2,3,8), (3,4,8), (4,5,8), (5,6,8), (6,7,8) )
gap> Size( last );
20160 ```

`GeneralLinearGroup( n, q )` `GeneralLinearGroup( D, n, q )`

In the first form `GeneralLinearGroup` returns the general linear group GL( <n>, <q> ) as a matrix group. In the second form D must be a domain of group elements, e.g., `Permutations` or `AgWords`, and `GeneralLinearGroup` returns GL( <n>, <q> ) as a group of elements of that type.

```    gap> g := GeneralLinearGroup( 2, 4 ); Size( g );
GL(2,4)
180 ```

`SpecialLinearGroup( n, q )` `SpecialLinearGroup( D, n, q )`

In the first form `SpecialLinearGroup` returns the special linear group SL( <n>, <q> ) as a matrix group. In the second form D must be a domain of group elements, e.g., `Permutations` or `AgWords`, and `SpecialLinearGroup` returns SL( <n>, <q> ) as a group of elements of that type.

```    gap> g := SpecialLinearGroup( 3, 4 ); Size( g );
SL(3,4)
60480 ```

`SymplecticGroup( n, q )` `SymplecticGroup( D, n, q )`

In the first form `SymplecticGroup` returns the symplectic group SP( <n>, <q> ) as a matrix group. In the second form D must be a domain of group elements, e.g., `Permutations` or `AgWords`, and `SymplecticGroup` returns SP( <n>, <q> ) as a group of elements of that type.

```    gap> g := SymplecticGroup( 4, 2 ); Size( g );
SP(4,2)
720 ```

`GeneralUnitaryGroup( n, q )` `GeneralUnitaryGroup( D, n, q )`

In the first form `GeneralUnitaryGroup` returns the general unitary group GU( <n>, <q> ) as a matrix group. In the second form D must be a domain of group elements, e.g., `Permutations` or `AgWords`, and `GeneralUnitaryGroup` returns GU( <n>, <q> ) as a group of elements of that type.

```    gap> g := GeneralUnitaryGroup( 3, 3 ); Size( g );
GU(3,3)
24192 ```

`SpecialUnitaryGroup( n, q )` `SpecialUnitaryGroup( D, n, q )`

In the first form `SpecialUnitaryGroup` returns the special unitary group SU( <n>, <q> ) as a matrix group. In the second form D must be a domain of group elements, e.g., `Permutations` or `AgWords`, and `SpecialUnitaryGroup` returns SU( <n>, <q> ) as a group of elements of that type.

```    gap> g := SpecialUnitaryGroup( 3, 3 ); Size( g );
SU(3,3)
6048 ```

`MathieuGroup( d )`

`MathieuGroup` returns the Mathieu group of degree d as a permutation group. d is expected to be 11, 12, 22, 23, or 24.

```    gap> g := MathieuGroup( 12 ); Size( g );
Group( ( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11), ( 3, 7,11, 8)
( 4,10, 5, 6), ( 1,12)( 2,11)( 3, 6)( 4, 8)( 5, 9)( 7,10) )
95040```

## 37.2 Selection Functions

`AllLibraryGroups( fun1, val1, fun2, val2, ... )`

For each group library there is a selection function. This function allows you to select all groups from the library that have a given set of properties.

The name of the selection functions always begins with `All` and always ends with `Groups`. Inbetween is a name that hints at the nature of the group library. For example, the selection function for the library of all The Primitive Groups Library) is called `AllPrimitiveGroups`, and the selection function for The 2-Groups Library) is called `AllTwoGroups`.

These functions take an arbitrary number of pairs of arguments. The first argument in such a pair is a function that can be applied to the groups in the library, and the second argument is either a single value that this function must return in order to have this group included in the selection, or a list of such values.

For example

```    AllPrimitiveGroups( DegreeOperation,  [10..15],
Size,             [1..100],
IsAbelian,        false    );```

should return a list of all primitive groups with degree between 10 and 15 and size less than 100 that are not abelian.

Thus the `AllPrimitiveGroups` behaves as if it was implemented by a function similar to the one defined below, where `PrimitiveGroupsList` is a list of all primitive groups. Note, in the definition below we assume for simplicity that `AllPrimitiveGroups` accepts exactly 4 arguments. It is of course obvious how to change this definition so that the function would accept a variable number of arguments.

```    AllPrimitiveGroups := function ( fun1, val1, fun2, val2 )
local    groups, g, i;
groups := [];
for i  in [ 1 .. Length( PrimitiveGroupsList ) ] do
g := PrimitiveGroupsList[i];
if      fun1(g) = val1  or IsList(val1) and fun1(g) in val1
and fun2(g) = val2  or IsList(val2) and fun2(g) in val2
then
fi;
od;
return groups;
end; ```

Note that the real selection functions are considerably more difficult, to improve the efficiency. Most important, each recognizes a certain set About Group Libraries).

## 37.3 Example Functions

`OneLibraryGroup( fun1, val1, fun2, val2, ... )`

For each group library there is a example function. This function allows you to find one group from the library that has a given set of properties.

The name of the example functions always begins with `One` and always ends with `Group`. Inbetween is a name that hints at the nature of the group library. For example, the example function for the library of all The Primitive Groups Library) is called `OnePrimitiveGroup`, and the example function for the library of all 2-groups of size at most 256 (see The 2-Groups Library) is called `OneTwoGroup`.

These functions take an arbitrary number of pairs of arguments. The first argument in such a pair is a function that can be applied to the groups in the library, and the second argument is either a single value that this function must return in order to have this group returned by the example function, or a list of such values.

For example

```    OnePrimitiveGroup( DegreeOperation,  [10..15],
Size,             [1..100],
IsAbelian,        false    );```

should return one primitive group with degree between 10 and 15 and size size less than 100 that is not abelian.

Thus the `OnePrimitiveGroup` behaves as if it was implemented by a function similar to the one defined below, where `PrimitiveGroupsList` is a list of all primitive groups. Note, in the definition below we assume for simplicity that `OnePrimitiveGroup` accepts exactly 4 arguments. It is of course obvious how to change this definition so that the function would accept a variable number of arguments.

```    OnePrimitiveGroup := function ( fun1, val1, fun2, val2 )
local    g, i;
for i  in [ 1 .. Length( PrimitiveGroupsList ) ] do
g := PrimitiveGroupsList[i];
if      fun1(g) = val1  or IsList(val1) and fun1(g) in val1
and fun2(g) = val2  or IsList(val2) and fun2(g) in val2
then
return g;
fi;
od;
return false;
end; ```

Note that the real example functions are considerably more difficult, to improve the efficiency. Most important, each recognizes a certain set of About Group Libraries).

## 37.4 Extraction Functions

For each group library there is an extraction function. This function allows you to extract single groups from the library.

The name of the extraction function always ends with `Group` and begins with a name that hints at the nature of the library. For example the The Primitive Groups Library) is called `PrimitiveGroup`, and the extraction The 2-Groups Library) is called `TwoGroup`.

What arguments the extraction function accepts, and how they are interpreted is described in the sections that describe the individual group libraries.

For example

` PrimitiveGroup( 10, 4 ); `

returns the 4-th primitive group of degree 10.

The reason for the extraction function is as follows. A group library is usually not stored as a list of groups. Instead a more compact representation for the groups is used. For example the groups in the library of 2-groups are represented by 4 integers. The extraction function hides this representation from you, and allows you to access the group library as if it was a table of groups (two dimensional in the above example).

## 37.5 The Primitive Groups Library

This group library contains all primitive permutation groups of degree at most 50. There are a total of 406 such groups. Actually to be a little bit more precise, there are 406 inequivalent primitive operations on at most 50 points. Quite a few of the 406 groups are isomorphic.

`AllPrimitiveGroups( fun1, val1, fun2, val2, ... )`

`AllPrimitiveGroups` returns a list containing all primitive groups that have the properties given as arguments. Each property is specified by passing a pair of arguments, the first being a function, which will be applied to all groups in the library, and the second being a value or a list of values, that this function must return in order to have this group included in the list returned by `AllPrimitiveGroups`.

The first argument must be `DegreeOperation` and the second argument either a degree or a list of degrees, otherwise `AllPrimitiveGroups` will print a warning to the effect that the library contains only groups with degrees between 1 and 50.

```    gap> l := AllPrimitiveGroups( Size, 120, IsSimple, false );
#W  AllPrimitiveGroups: degree automatically restricted to [1..50]
[ S(5), PGL(2,5), S(5) ]
gap> List( l, g -> g.generators );
[ [ (1,2,3,4,5), (1,2) ], [ (1,2,3,4,5), (2,3,5,4), (1,6)(3,4) ],
[ ( 1, 8)( 2, 5, 6, 3)( 4, 9, 7,10), ( 1, 5, 7)( 2, 9, 4)( 3, 8,10)
] ] ```

`OnePrimitiveGroup( fun1, val1, fun2, val2, ... )`

`OnePrimitiveGroup` returns one primitive group that has the properties given as argument. Each property is specified by passing a pair of arguments, the first being a function, which will be applied to all groups in the library, and the second being a value or a list of values, that this function must return in order to have this group returned by `OnePrimitiveGroup`. If no such group exists, `false` is returned.

The first argument must be `DegreeOperation` and the second argument either a degree or a list of degrees, otherwise `OnePrimitiveGroup` will print a warning to the effect that the library contains only groups with degrees between 1 and 50.

```    gap> g := OnePrimitiveGroup( DegreeOperation,5, IsSolvable,false );
A(5)
gap> Size( g );
60 ```

`AllPrimitiveGroups` and `OnePrimitiveGroup` recognize the following functions and handle them usually quite efficient. `DegreeOperation`, `Size`, `Transitivity`, and `IsSimple`. You should pass those functions first, e.g., it is more efficient to say ```AllPrimitiveGroups( Size,120 , IsAbelian,false )``` than to say ```AllPrimitiveGroups( IsAbelian,false, Size,120 )``` (see About Group Libraries).

`PrimitiveGroup( deg, nr )`

`PrimitiveGroup` returns the nr-th primitive group of degree deg. Both deg and nr must be positive integers. The primitive groups of equal degree are sorted with respect to their size, so for example `PrimitiveGroup( deg, 1 )` is the smallest primitive group of degree deg, e.g, the cyclic group of size deg, if deg is a prime. Primitive groups of equal degree and size are in no particular order.

```    gap>  g := PrimitiveGroup( 8, 1 );
AGL(1,8)
gap> g.generators;
[ (1,2,3,4,5,6,7), (1,8)(2,4)(3,7)(5,6) ] ```

Apart from the usual components described in Group Records, the group records returned by the above functions have the following components.

`transitivity`:

degree of transitivity of G.

`isSharpTransitive`:

`true` if G is sharply `G.transitivity`-fold transitive and `false` otherwise.

`isKPrimitive`:

`true` if G is k-fold primitive, and `false` otherwise.

`isOdd`:

`false` if G is a subgroup of the alternating group of degree `G.degree` and `true` otherwise.

`isFrobeniusGroup`:

`true` if G is a Frobenius group indexFrobenius group and `false` otherwise.

This library was computed by Charles Sims. The list of primitive permutation groups of degree at most 20 was published in Sim70. The library was brought into GAP format by Martin Schaccent127onert. He assumes the responsibility for all mistakes.

## 37.6 The Transitive Groups Library

The transitive groups library contains representatives for all transitive permutation groups of degree at most 22. Two permutations groups of the same degree are considered to be equivalent, if there is a renumbering of points, which maps one group into the other one. In other words, if they lie in the save conjugacy class under operation of the full symmetric group by conjugation.

There are a total of 4945 such groups up to degree 22.

`AllTransitiveGroups( fun1, val1, fun2, val2, ... )`

`AllTransitiveGroups` returns a list containing all transitive groups that have the properties given as arguments. Each property is specified by passing a pair of arguments, the first being a function, and the second being a value or a list of values. `AllTransitiveGroups` will return all groups from the transitive groups library, for which all specified functions have the specified values.

If the degree is not restricted to 22 at most, `AllTransitiveGroups` will print a warning.

`OneTransitiveGroup( fun1, val1, fun2, val2, ... )`

`OneTransitiveGroup` returns one transitive group that has the properties given as argument. Each property is specified by passing a pair of arguments, the first being a function, and the second being a value or a list of values. `OneTransitiveGroup` will return one groups from the transitive groups library, for which all specified functions have the specified values. If no such group exists, `false` is returned.

If the degree is not restricted to 22 at most, `OneTransitiveGroup` will print a warning.

`AllTransitiveGroups` and `OneTransitiveGroup` recognize the following functions and get the corresponding properties from a precomputed list to speed up processing:
`DegreeOperation`, `Size`, `Transitivity`, and `IsPrimitive`. You do not need to pass those functions first, as the selection function picks the these properties first.

`TransitiveGroup( deg, nr )`

`TransitiveGroup` returns the nr-th transitive group of degree deg. Both deg and nr must be positive integers. The transitive groups of equal degree are sorted with respect to their size, so for example `TransitiveGroup( deg, 1 )` is the smallest transitive group of degree deg, e.g, the cyclic group of size deg, if deg is a prime. The ordering of the groups corresponds to the list in Butler/McKay BM83.

This library was computed by Gregory Butler, John McKay, Gordon Royle and Alexander Hulpke. The list of transitive groups up to degree 11 was published in BM83, the list of degree 12 was published in Roy87, degree 14 and 15 were published in But93.

The library was brought into GAP format by Alexander Hulpke, who is responsible for all mistakes.

```    gap> TransitiveGroup(10,22);
S(5)[x]2
gap> l:=AllTransitiveGroups(DegreeOperation,12,Size,1440,
> IsSolvable,false);
[ S(6)[x]2, M_10.2(12) = A_6.E_4(12) = [S_6[1/720]{M_10}S_6]2 ]
gap> List(l,IsSolvable);
[ false, false ] ```

`TransitiveIdentification( G )`

Let G be a permutation group, acting transitively on a set of up to 22 points. Then `TransitiveIdentification` will return the position of this group in the transitive groups library. This means, if G operates on m points and `TransitiveIdentification` returns n, then G is permutation isomorphic to the group `TransitiveGroup(m,n)`.

```    gap> TransitiveIdentification(Group((1,2),(1,2,3)));
2 ```

## 37.7 The Solvable Groups Library

GAP has a library of the 1045 solvable groups of size between 2 and 100. The groups are from lists computed by M.~Hall and J.~K.~Senior (size 64, see HS64), R.~Laue (size 96, see Lau82) and J.~Neubaccent127user (other sizes, see Neu67).

`AllSolvableGroups( fun1, val1, fun2, val2, ... )`

`AllSolvableGroups` returns a list containing all solvable groups that have the properties given as arguments. Each property is specified by passing a pair of arguments, the first being a function, which will be applied to all the groups in the library, and the second being a value or a list of values, that this function must return in order to have this group included in the list returned by `AllSolvableGroups`.

```    gap> AllSolvableGroups(Size,24,IsNontrivialDirectProduct,false);
[ 12.2, grp_24_11, D24, Q8+S3, Sl(2,3), S4 ] ```

`OneSolvableGroup( fun1, val1, fun2, val2, ... )`

`OneSolvableGroup` returns a solvable group with the specified properties. Each property is specified by passing a pair of arguments, the first being a function, which will be applied to all the groups in the library, and the second being a value or a list of values, that this function must return in order to have this group returned by `OneSolvableGroup`. If no such group exists, `false` is returned.

```    gap> OneSolvableGroup(Size,100,x->Size(DerivedSubgroup(x)),10);
false
gap> OneSolvableGroup(Size,24,IsNilpotent,false);
S3x2^2 ```

`AllSolvableGroups` and `OneSolvableGroup` recognize the following functions and handle them usually very efficiently: `Size`, `IsAbelian`, `IsNilpotent`, and `IsNonTrivialDirectProduct`.

`SolvableGroup( size, nr )`

`SolvableGroup` returns the nr-th group of size size in the library. `SolvableGroup` will signal an error if size is not between 2 and 100, or if nr is larger than the number of solvable groups of size size. Finite Polycyclic Groups).

```    gap> SolvableGroup( 32 , 15 );
Q8x4 ```

## 37.8 The 2-Groups Library

The library of 2-groups contains all the 2-groups of size dividing 256. There are a total of 58760 such groups, 1 of size 2, 2 of size 4, 5 of size 8, 14 of size 16, 51 of size 32, 267 of size 64, 2328 of size 128, and 56092 of size 256.

`AllTwoGroups( fun1, val1, fun2, val2, ... )`

`AllTwoGroups` returns the list of all the 2-groups that have the properties given as arguments. Each property is specified by passing a pair of arguments, the first is a function that can be applied to each group, the second is either a single value or a list of values that the function must return in order to select that group.

```    gap> l := AllTwoGroups( Size, 256, Rank, 3, pClass, 2 );
[ Group( a1, a2, a3, a4, a5, a6, a7, a8 ),
Group( a1, a2, a3, a4, a5, a6, a7, a8 ),
Group( a1, a2, a3, a4, a5, a6, a7, a8 ),
Group( a1, a2, a3, a4, a5, a6, a7, a8 ) ]
gap> List( l, g -> Length( ConjugacyClasses( g ) ) );
[ 112, 88, 88, 88 ] ```

`OneTwoGroup( fun1, val1, fun2, val2, ... )`

`OneTwoGroup` returns a single 2-group that has the properties given as arguments. Each property is specified by passing a pair of arguments, the first is a function that can be applied to each group, the second is either a single value or a list of values that the function must return in order to select that group.

```    gap> g := OneTwoGroup( Size, [64..128], Rank, [2..3], pClass, 5 );
#I  size restricted to [ 64, 128 ]
Group( a1, a2, a3, a4, a5, a6 )
gap> Size( g );
64
gap> Rank( g );
2 ```

`AllTwoGroups` and `OneTwoGroup` recognize the following functions and handle them usually very efficiently. `Size`, `Rank` for the rank of the Frattini quotient of the group, and `pClass` for the exponent-p class of the group. Note that `Rank` and `pClass` are dummy functions that can be used only in this context, i.e., they can not be applied to arbitrary groups.

`TwoGroup( size, nr )`

`TwoGroup` returns the nr-th group of size size. The group is returned as a finite polycyclic group (see Finite Polycyclic Groups). `TwoGroup` will signal an error if size is not a power of 2 between 2 and 256, or nr is larger than the number of groups of size size.

Within each size the following criteria have been used, in turn, to determine the index position of a group in the list

1:
increasing generator number;

2:
increasing exponent-2 class;

3:
the position of its parent in the list of groups of appropriate size;

4:
the list in which the Newman and O'Brien implementation of the p-group generation algorithm outputs the immediate descendants of a group.

```    gap> g := TwoGroup( 32, 45 );
Group( a1, a2, a3, a4, a5 )
gap> Rank( g );
4
gap> pClass( g );
2
gap> g.abstractRelators;
[ a1^2*a5^-1, a2^2, a2^-1*a1^-1*a2*a1, a3^2, a3^-1*a1^-1*a3*a1,
a3^-1*a2^-1*a3*a2, a4^2, a4^-1*a1^-1*a4*a1, a4^-1*a2^-1*a4*a2,
a4^-1*a3^-1*a4*a3, a5^2, a5^-1*a1^-1*a5*a1, a5^-1*a2^-1*a5*a2,
a5^-1*a3^-1*a5*a3, a5^-1*a4^-1*a5*a4 ] ```

Apart from the usual components described in Group Records, the group records returned by the above functions have the following components.

`rank`:

rank of Frattini quotient of G.

`pclass`:

exponent-p class of G.

`abstractGenerators`:

a list of abstract generators of G (see AbstractGenerator).

`abstractRelators`:

a list of relators of G stored as words in the abstract generators.

Descriptions of the algorithms used in constructing the library data may be found in citeOBr90,OBr91. Using these techniques, a library was first prepared in 1987 by M.F.~Newman and E.A.~O'Brien; a partial description may be found in NO89.

The library was brought into the GAP format by Werner Nickel, Alice Niemeyer, and E.A.~O'Brien.

## 37.9 The 3-Groups Library

The library of 3-groups contains all the 3-groups of size dividing 729. There are a total of 594 such groups, 1 of size 3, 2 of size 9, 5 of size 27, 15 of size 81, 67 of size 243, and 504 of size 729.

`AllThreeGroups( fun1, val1, fun2, val2, ... )`

`AllThreeGroups` returns the list of all the 3-groups that have the properties given as arguments. Each property is specified by passing a pair of arguments, the first is a function that can be applied to each group, the second is either a single value or a list of values that the function must return in order to select that group.

```    gap> l := AllThreeGroups( Size, 243, Rank, [2..4], pClass, 3 );;
gap> Length ( l );
33
gap>  List( l, g -> Length( ConjugacyClasses( g ) ) );
[ 35, 35, 35, 35, 35, 35, 35, 243, 99, 99, 51, 51, 51, 51, 51, 51,
51, 51, 99, 35, 243, 99, 99, 51, 51, 51, 51, 51, 35, 35, 35, 35, 35
] ```

`OneThreeGroup( fun1, val1, fun2, val2, ... )`

`OneThreeGroup` returns a single 3-group that has the properties given as arguments. Each property is specified by passing a pair of arguments, the first is a function that can be applied to each group, the second is either a single value or a list of values that the function must return in order to select that group.

```    gap> g := OneThreeGroup( Size, 729, Rank, 4, pClass, [3..5] );
Group( a1, a2, a3, a4, a5, a6 )
gap> IsAbelian( g );
true ```

`AllThreeGroups` and `OneThreeGroup` recognize the following functions and handle them usually very efficiently. `Size`, `Rank` for the rank of the Frattini quotient of the group, and `pClass` for the exponent-p class of the group. Note that `Rank` and `pClass` are dummy functions that can be used only in this context, i.e., they cannot be applied to arbitrary groups.

`ThreeGroup( size, nr )`

`ThreeGroup` returns the nr-th group of size size. The group is returned as a finite polycyclic group (see Finite Polycyclic Groups). `ThreeGroup` will signal an error if size is not a power of 3 between 3 and 729, or nr is larger than the number of groups of size size.

Within each size the following criteria have been used, in turn, to determine the index position of a group in the list

1:
increasing generator number;

2:
increasing exponent-3 class;

3:
the position of its parent in the list of groups of appropriate size;

4:
the list in which the Newman and O'Brien implementation of the p-group generation algorithm outputs the immediate descendants of a group.

```    gap> g := ThreeGroup( 243, 56 );
Group( a1, a2, a3, a4, a5 )
gap> pClass( g );
3
gap> g.abstractRelators;
[ a1^3, a2^3, a2^-1*a1^-1*a2*a1*a4^-1, a3^3, a3^-1*a1^-1*a3*a1,
a3^-1*a2^-1*a3*a2*a5^-1, a4^3, a4^-1*a1^-1*a4*a1*a5^-1,
a4^-1*a2^-1*a4*a2, a4^-1*a3^-1*a4*a3, a5^3, a5^-1*a1^-1*a5*a1,
a5^-1*a2^-1*a5*a2, a5^-1*a3^-1*a5*a3, a5^-1*a4^-1*a5*a4 ] ```

Apart from the usual components described in Group Records, the group records returned by the above functions have the following components.

`rank`:

rank of Frattini quotient of G.

`pclass`:

exponent-p class of G.

`abstractGenerators`:

a list of abstract generators of G (see AbstractGenerator).

`abstractRelators`:

a list of relators of G stored as words in the abstract generators.

Descriptions of the algorithms used in constructing the library data may be found in citeOBr90,OBr91.

The library was generated and brought into GAP format by E.A.~O'Brien and Colin Rhodes. David Baldwin, M.F.~Newman, and Maris Ozols have contributed in various ways to this project and to correctly determining these groups. The library design is modelled on and borrows extensively from the 2-groups library, which was brought into GAP format by Werner Nickel, Alice Niemeyer, and E.A.~O'Brien.

## 37.10 The Irreducible Solvable Linear Groups Library

The IrredSol group library provides access to the irreducible solvable subgroups of GL(n,p), where n > 1, p is prime and p^n < 256. The library contains exactly one member from each of the 370 conjugacy classes of such subgroups.

By well known theory, this library also doubles as a library of primitive solvable permutation groups of non-prime degree less than 256. To access the data in this form, you must first build the matrix group(s) of interest and then call the function
`PrimitivePermGroupIrreducibleMatGroup( matgrp )`
This function returns a permutation group isomorphic to the semidirect product of an irreducible matrix group (over a finite field) and its underlying vector space.

`AllIrreducibleSolvableGroups( fun1, val1, fun2, val2, ... )`

`AllIrreducibleSolvableGroups` returns a list containing all irreducible solvable linear groups that have the properties given as arguments. Each property is specified by passing a pair of arguments, the first being a function which will be applied to all groups in the library, and the second being a value or a list of values that this function must return in order to have this group included in the list returned by `AllIrreducibleSolvableGroups`.

```    gap> AllIrreducibleSolvableGroups( Dimension, 2,
>                                   CharFFE, 3,
>                                   Size, 8 );
[ Group( [ [ 0*Z(3), Z(3)^0 ], [ Z(3)^0, 0*Z(3) ] ],
[ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0 ] ],
[ [ Z(3)^0, 0*Z(3) ], [ 0*Z(3), Z(3) ] ] ),
Group( [ [ 0*Z(3), Z(3)^0 ], [ Z(3), 0*Z(3) ] ],
[ [ Z(3)^0, Z(3) ], [ Z(3), Z(3) ] ] ),
Group( [ [ 0*Z(3), Z(3)^0 ], [ Z(3)^0, Z(3) ] ] ) ] ```

`OneIrreducibleSolvableGroup( fun1, val1, fun2, val2, ... )`

`OneIrreducibleSolvableGroup` returns one irreducible solvable linear group that has the properties given as arguments. Each property is specified by passing a pair of arguments, the first being a function which will be applied to all groups in the library, and the second being a value or a list of values that this function must return in order to have this group returned by `OneIrreducibleSolvableGroup`. If no such group exists, `false` is returned.

```    gap> OneIrreducibleSolvableGroup( Dimension, 4,
>                                  IsLinearlyPrimitive, false );
Group( [ [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
[ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ] ],
[ [ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
[ Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ] ],
[ [ Z(2)^0, 0*Z(2), 0*Z(2), 0*Z(2) ],
[ 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ],
[ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ],
[ 0*Z(2), 0*Z(2), Z(2)^0, Z(2)^0 ] ] ) ```

`AllIrreducibleSolvableGroups` and `OneIrreducibleSolvableGroup` recognize the following functions and handle them very efficiently (because the information is stored with the groups and so no computations are needed): `Dimension` for the linear degree, `CharFFE` for the field characteristic, `Size`, `IsLinearlyPrimitive`, and `MinimalBlockDimension`. Note that the last two are dummy functions that can be used only in this context. Their meaning is explained at the end of this section.

`IrreducibleSolvableGroup( n, p, i )`

`IrreducibleSolvableGroup` returns the i-th irreducible solvable subgroup of GL( n, p ). The irreducible solvable subgroups of

GL(n,p) are ordered with respect to the following criteria:
item increasing size; item increasing guardian number. If two groups have the same size and guardian, they are in no particular order. (See the library documentation or [Sho92] for the meaning of guardian.)

```    gap> g := IrreducibleSolvableGroup( 3, 5, 12 );
Group( [ [ 0*Z(5), Z(5)^2, 0*Z(5) ], [ Z(5)^2, 0*Z(5), 0*Z(5) ],
[ 0*Z(5), 0*Z(5), Z(5)^2 ] ],
[ [ 0*Z(5), Z(5)^0, 0*Z(5) ], [ 0*Z(5), 0*Z(5), Z(5)^0 ],
[ Z(5)^0, 0*Z(5), 0*Z(5) ] ],
[ [ Z(5)^2, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^0, 0*Z(5) ],
[ 0*Z(5), 0*Z(5), Z(5)^2 ] ],
[ [ Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^2, 0*Z(5) ],
[ 0*Z(5), 0*Z(5), Z(5)^2 ] ],
[ [ Z(5), 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5), 0*Z(5) ],
[ 0*Z(5), 0*Z(5), Z(5) ] ] ) ```

Apart from the usual components described in Group Records, the group records returned by the above functions have the following components.

`size`:

size of G.

`isLinearlyPrimitive`:

`false` if G preserves a direct sum decomposition of the underlying vector space, and `true` otherwise.

`minimalBlockDimension`:

not bound if G is linearly primitive; otherwise equals the dimension of the blocks in an unrefinable system of imprimitivity for G.

This library was computed and brought into GAP format by Mark Short. Descriptions of the algorithms used in computing the library data can be found in Sho92.

## 37.11 The Library of Finite Perfect Groups

The GAP library of finite perfect groups provides, up to isomorphism, a list of all perfect groups whose sizes are less than 10^6 excluding the following: item For n = 61440, 122880, 172032, 245760, 344064, 491520, 688128, or 983040, the perfect groups of size n have not completely been determined yet. The library neither provides the number of these groups nor the groups themselves. vspace-1mm item For n = 86016, 368640, or 737280, the library does not yet contain the perfect groups of size n, it only provides their their numbers which are 52, 46, or 54, respectively. vspace-2mm

Except for these eleven sizes, the list of altogether 1096 perfect groups in the library is complete. It relies on results of Derek F.~Holt and Wilhelm Plesken which are published in their book it Perfect Groups HP89. Moreover, they have supplied to us files with presentations of 488 of the groups. In terms of these, the remaining 607 nontrivial groups in the library can be described as 276 direct products, 107 central products, and 224 subdirect products. They are computed automatically by suitable GAP functions whenever they are needed.

We are grateful to Derek Holt and Wilhelm Plesken for making their groups available to the GAP community by contributing their files. It should be noted that their book contains a lot of further information for many of the library groups. So we would like to recommend it to any GAP user who is interested in the groups.

The library has been brought into GAP format by Volkmar Felsch.

Like most of the other GAP libraries, the library of finite perfect groups provides an extraction function, `PerfectGroup`. It returns the specified group in form of a finitely presented group which, in its group record, bears some additional information that allows you to easily construct an isomorphic permutation group of some appropriate degree by just calling the `PermGroup` function.

Further, there is a function `NumberPerfectGroups` which returns the number of perfect groups of a given size.

The library does not provide a selection or an example function. There is, however, a function `DisplayInformationPerfectGroups` which allows the display of some information about arbitrary library groups without actually loading the large files with their presentations, and without constructing the groups themselves.

Moereover, there are two functions which allow you to formulate loops over selected library groups. The first one is the `NumberPerfectLibraryGroups` function which, for any given size, returns the number of groups in the library which are of that size.

The second one is the `SizeNumbersPerfectGroups` function. It allows you to ask for all library groups which contain certain composition factors. The answer is provided in form of a list of pairs [size,n] where each such pair characterizes the n^{rm th} library group of size size. We will call such a pair [size,n] the size number of the respective perfect group. As the size numbers are accepted as input arguments by the `PerfectGroup` and the `DisplayInformationPerfectGroups` function, you may use their list to formulate a loop over the associated groups.

Now we shall give an individual description of each library function.

`NumberPerfectGroups( size )`

`NumberPerfectGroups` returns the number of non-isomorphic perfect groups of size size for each positive integer size up to 10^6 except for the eight sizes listed at the beginning of this section for which the number is not yet known. For these values as well as for any argument out of range it returns the value -1.

`NumberPerfectLibraryGroups( size )`

`NumberPerfectLibraryGroups` returns the number of perfect groups of size size which are available in the library of finite perfect groups.

The purpose of the function is to provide a simple way to formulate a loop over all library groups of a given size.

`SizeNumbersPerfectGroups( factor1, factor2 ... )`

`SizeNumbersPerfectGroups` returns a list of the it size numbers (see above) of all library groups that contain the specified factors among their composition factors. Each argument must either be the name of a simple group or an integer expression which is the product of the sizes of one or more cyclic factors. The function ignores the order in which the argmuments are given and, in fact, replaces any list of more than one integer expression among the arguments by their product.

The following text strings are accepted as simple group names. item[] `"A5"`, `"A6"`, `"A7"`, `"A8"`, `"A9"` or `"A(5)"`, `"A(6)"`, `"A(7)"`, `"A(8)"`, `"A(9)"` for the alternating groups A_n, 5 leq n leq 9, vspace-2mm item[] `"L2(`q`)"` or `"L(2,`q`)"` for PSL(2,q), where q is any prime power with 4 leq q leq 125, vspace-2mm item[] `"L3(`q`)"` or `"L(3,`q`)"` for PSL(3,q) with 2 leq q leq 5, vspace-2mm item[] `"U3(`q`)"` or `"U(3,`q`)"` for PSU(2,q) with 3 leq q leq 5, vspace-2mm item[] `"U4(2)` or `"U(4,2)"` for PSU(4,2), vspace-2mm item[] `"Sp4(4)"` or `"S(4,4)"` for the symplectic group S(4,4), vspace-2mm item[] `"Sz(8)"` for the Suzuki group Sz(8), vspace-2mm item[] `"M11"`, `"M12"`, `"M22"` or `"M(11)"`, `"M(12)"`, `"M(22)"` for the Matthieu groups M_{11}, M_{12}, and M_{22}, and vspace-2mm item[] `"J1"`, `"J2"` or `"J(1)"`, `"J(2)"` for the Janko groups J_1 and J_2. vspace-2mm

Note that, for most of the groups, the preceding list offers two different names in order to be consistent with the notation used in HP89 as well as with the notation used in the `DisplayCompositionSeries` command of GAP. However, as the names are compared as text strings, you are restricted to the above choice. Even expressions like `"L2(`~`32`~`)"` or `"L2(2^5)"` are not accepted.

As the use of the term PSU(n,q) is not unique in the literature, we state that here it denotes the factor group of SU(n,q) by its centre, where SU(n,q) is the group of all n times n unitary matrices with entries in GF(q^2) and determinant 1.

The purpose of the function is to provide a simple way to formulate a loop over all library groups which contain certain composition factors.

`DisplayInformationPerfectGroups( size )` `DisplayInformationPerfectGroups( size, n )`
`DisplayInformationPerfectGroups( [ size, n ] )`

`DisplayInformationPerfectGroups` displays some information about the library group G, say, which is specified by the size number [size,n] or by the two arguments size and n. If, in the second case, n is omitted, the function will loop over all library groups of size size.

The information provided for G includes the following items: item a headline containing the size number [size,n] of G in the form size.n (the suffix .n will be suppressed if, up to isomorphism, G is the only perfect group of size size), vspace-2mm item a message if G is simple or quasisimple, i.,e., if the factor group of G by its centre is simple, vspace-2mm item the ``description'' of the structure of G as it is given by Holt and Plesken in HP89 (see below), vspace-2mm item the size of the centre of G (suppressed, if G is simple), vspace-2mm item the prime decomposition of the size of G, vspace-2mm item orbit sizes for a faithful permutation representation of G which is provided by the library (see below), vspace-2mm item a reference to each occurrence of G in the tables of section 5.3 of HP89. Each of these references consists of a class number and an internal number (i,j) under which G is listed in that class. For some groups, there is more than one reference because these groups belong to more than one of the classes in the book. vspace-2mm Example:

```    gap> DisplayInformationPerfectGroups( 30720, 3 );
#I Perfect group 30720.3:  A5 ( 2^4 E N 2^1 E 2^4 ) A
#I   centre = 1  size = 2^11*3*5  orbit size = 240
#I   Holt-Plesken class 1 (9,3)
gap> DisplayInformationPerfectGroups( 30720, 6 );
#I Perfect group 30720.6:  A5 ( 2^4 x 2^4 ) C N 2^1
#I   centre = 2  size = 2^11*3*5  orbit size = 384
#I   Holt-Plesken class 1 (9,6)
gap> DisplayInformationPerfectGroups( Factorial( 8 ) / 2 );
#I Perfect group 20160.1:  A5 x L3(2) 2^1
#I   centre = 2  size = 2^6*3^2*5*7  orbit sizes = 5 + 16
#I   Holt-Plesken class 31 (1,1) (occurs also in class 32)
#I Perfect group 20160.2:  A5 2^1 x L3(2)
#I   centre = 2  size = 2^6*3^2*5*7  orbit sizes = 7 + 24
#I   Holt-Plesken class 31 (1,2) (occurs also in class 32)
#I Perfect group 20160.3:  ( A5 x L3(2) ) 2^1
#I   centre = 2  size = 2^6*3^2*5*7  orbit size = 192
#I   Holt-Plesken class 31 (1,3)
#I Perfect group 20160.4:  simple group  A8
#I   size = 2^6*3^2*5*7  orbit size = 8
#I   Holt-Plesken class 26 (0,1)
#I Perfect group 20160.5:  simple group  L3(4)
#I   size = 2^6*3^2*5*7  orbit size = 21
#I   Holt-Plesken class 27 (0,1) ```

For any library group G, the library files do not only provide a presentation, but, in addition, a list of one or more subgroups S_1, ldots, S_r of G such that there is a faithful permutation representation of G of degree sum_{i=1}^{r} G !:! S_i on the set { S_i g mid 1 leq i leq r, , g in G } of the cosets of the S_i. The respective permutation group is available via the `PermGroup` function described below. The `DisplayInformationPerfectGroups` function displays only the available degree. The message

` orbit size = 8 `

in the above example means that the available permutation representation is transitive and of degree 8, whereas the message

` orbit sizes = 7 + 24 `

means that a nontransitive permutation representation is available which acts on two orbits of size 7 and 24 respectively.

The notation used in the ``description'' of a group is explained in section 5.1.2 of HP89. We quote the respective page from there:

sl `Within a class Q,#,p, an isomorphism type of groups will be denoted by an ordered pair of integers (r,n), where r geq 0 and n > 0. More precisely, the isomorphism types in Q # p of order p^r !! mid !! Q !! mid will be denoted by (r,1), (r,2), (r,3), ldots,. Thus Q will always get the size number (0,1).

In addition to the symbol (r,n), the groups in Q , {sl #} , p will also be given a more descriptive name. The purpose of this is to provide a very rough idea of the structure of the group. The names are derived in the following manner. First of all, the isomorphism classes of irreducible F_pQ-modules M with mid !! Q !! mid mid !! M !! mid , leq 10^6, where F_p is the field of order p, are assigned symbols. These will either be simply p^x, where x is the dimension of the module, or, if there is more than one isomorphism class of irreducible modules having the same dimension, they will be denoted by p^x, p^{x^prime}, etc. The one-dimensional module with trivial Q-action will therefore be denoted by p^1. These symbols will be listed under the description of Q. The group name consists essentially of a list of the composition factors working from the top of the group downwards; hence it always starts with the name of Q itself. (This convention is the most convenient in our context, but it is different from that adopted in the ATLAS (Conway it et al.~1985), for example, where composition factors are listed in the reverse order. For example, we denote a group isomorphic to SL(2,5) by A_5 2^1 rather than 2 cdot A_5.)

Some other symbols are used in the name, in order to give some idea of the relationship between these composition factors, and splitting properties. We shall now list these additional symbols. item[times] between two factors denotes a direct product of F_pQ-modules or groups. vspace-2mm item[C] (for `commutator') between two factors means that the second lies in the commutator subgroup of the first. Similarly, a segment of the form (f_1 ! times ! f_2) {sl C} f_3 would mean that the factors f_1 and f_2 commute modulo f_3 and f_3 lies in [f_1,f_2]. vspace-2mm item[A] (for `abelian') between two factors indicates that the second is in the pth power (but not the commutator subgroup) of the first. `A' may also follow the factors, if bracketed. vspace-7mm item[E] (for `elementary abelian') between two factors indicates that together they generate an elementary abelian group (modulo subsequent factors), but that the resulting F_pQ-module extension does not split. vspace-2mm item[N] (for `nonsplit') before a factor indicates that Q (or possibly its covering group) splits down as far at this factor but not over the factor itself. So `Q f_1 {sl N} f_2' means that the normal subgroup f_1f_2 of the group has no complement but, modulo f_2, f_1, does have a complement. vspace-2mm

Brackets have their obvious meaning. Summarizing, we have:
item[times] = dirext product; vspace-2mm item[C] = commutator subgroup; vspace-2mm item[A] = abelian; vspace-2mm item[E] = elementary abelian; and vspace-2mm item[N] = nonsplit. vspace-2mm Here are some examples. item[(i)] A_5 (2^4 {sl E} 2^1 {sl E} 2^4) {sl A} means that the pairs 2^4 {sl E} 2^1 and 2^1 {sl E} 2^4 are both elementary abelian of exponent 4. vspace-1mm item[(ii)] A_5 (2^4 {sl E} 2^1 {sl A}) {sl C} 2^1 means that O_2(G) is of symplectic type 2^{1+5}, with Frattini factor group of type 2^4 {sl E} 2^1. The `A' after the 2^1 indicates that G has a central cyclic subgroup 2^1 {sl A} 2^1 of order 4. vspace-1mm item[(iii)] L_3(2) ((2^1 {sl E}) ! times ! ({sl N} 2^3 {sl E} 2^{3^prime} {sl A}) {sl C}) 2^{3^prime} means that the 2^{3^prime} factor at the bottom lies in the commutator subgroup of the pair 2^3 {sl E} 2^{3^prime} in the middle, but the lower pair 2^{3^prime} {sl A} 2^{3^prime} is abelian of exponent 4. There is also a submodule 2^1 {sl E} 2^{3^prime}, and the covering group L_3(2) 2^1 of L_3(2) does not split over the 2^3 factor. (Since G is perfect, it goes without saying that the extension L_3(2) 2^1 cannot split itself.) vspace-2mm

We must stress that this notation does not always succeed in being precise or even unambiguous, and the reader is free to ignore it if it does not seem helpful.'

If such a group description has been given in the book for G (and, in fact, this is the case for most of the library groups), it is displayed by the `DisplayInformationPerfectGroups` function. Otherwise the function provides a less explicit description of the (in these cases unique) Holt-Plesken class to which G belongs, together with a serial number if this is necessary to make it unique.

`PerfectGroup( size )` `PerfectGroup( size, n )`
`PerfectGroup( [ size, n ] )`

`PerfectGroup` is the essential extraction function of the library. It returns a finitely presented group, G say, which is isomorphic to the library group specified by the size number [size,n] or by the two separate arguments size and n. In the second case, you may omit the parameter n. Then the default value is n = 1.

```    gap> G := PerfectGroup( 6048 );
PerfectGroup(6048)
gap> G.generators;
[ a, b ]
gap> G.relators;
[ a^2, b^6, a*b*a*b*a*b*a*b*a*b*a*b*a*b,
a*b^2*a*b^2*a*b^2*a*b^-2*a*b^-2*a*b^-2,
a*b*a*b^-2*a*b*a*b^-2*a*b*a*b^-2*a*b*a*b^-1*a*b^-1 ]
gap> G.size;
6048
gap> G.description;
"U3(3)"
gap> G.subgroups;
[ Subgroup( PerfectGroup(6048), [ a, b*a*b*a*b*a*b^3 ] ) ] ```

The generators and relators of G coincide with those given in HP89.

Note that, besides the components that are usually initialized for any finitely presented group, the group record of G contains the following components:

`size`:

the size of G,

`isPerfect`:

always `true`,

`description`:

the description of G as described with the `DisplayInformationPerfectGroups` function above,

`source`:

some internal information used by the library functions,

`subgroups`:

a list of subgroups S_1, ldots, S_r of G such that G acts faithfully on on the union of the sets of all cosets of the S_i.

The last of these components exists only if G is one of the 488 nontrivial library groups which are given directly by a presentation on file, i.,e., which are not constructed from other library groups in form of a direct, central, or subdirect product. It will be required by the following function.

`PermGroup( G )`

`PermGroup` returns a permutation group, P say, which is isomorphic to the given group G which is assumed to be a finitely presented perfect group that has been extracted from the library of finite perfect groups via the `PerfectGroup` function.

Let S_1, ldots, S_r be the subgroups listed in the component `G.subgroups` of the group record of G. Then the resulting group P is the permutation group of degree sum_{i=1}^{r} G !:! S_i which is induced by G on the set { S_i g mid 1 leq i leq r, g in G } of all cosets of the S_i.

Example (continued):

```    gap> P := PermGroup( G );
PermGroup(PerfectGroup(6048))
gap> P.size;
6048
gap> P.degree;
28 ```

Note that some of the library groups do not have a faithful permutation representation of small degree. Computations in these groups may be rather time consuming.

Example:

```    gap> P := PermGroup( PerfectGroup( 129024, 2 ) );
PermGroup(PerfectGroup(129024,2))
gap> P.degree;
14336 ```

## 37.12 Irreducible Maximal Finite Integral Matrix Groups

A library of irreducible maximal finite integral matrix groups is provided with GAP. It contains Q-class representatives for all of these groups of dimension at most 24, and Z-class representatives for those of dimension at most 11 or of dimension 13, 17, 19, or 23.

The groups provided in this library have been determined by Wilhelm Plesken, partially as joint work with Michael Pohst, or by members of his institute (Lehrstuhl B faccent127ur Mathematik, RWTH Aachen). In particular, the data for the groups of dimensions 2 to 9 have been taken from the output of computer calculations which they performed in 1979 (see PP77, PP80). The Z-class representatives of the groups of dimension 10 have been determined and computed by Bernd Souvignier (Sou94), and those of dimensions 11, 13, and 17 have been recomputed for this library from the circulant Gram matrices given in Ple85, using the stand-alone programs for the computation of short vectors and Bravais groups which have been developed in Plesken's institute. The Z-class representatives of the groups of dimensions 19 and 23 had already been determined in Ple85. Gabriele Nebe has recomputed them for us. Her main contribution to this library, however, is that she has determined and computed the Q-class representatives of the groups of non-prime dimensions between 12 and 24 (see PN95, NP95, Neb95).

The library has been brought into GAP format by Volkmar Felsch. He has applied several GAP routines to check certain consistency of the data. However, the credit and responsibility for the lists remain with the authors. We are grateful to Wilhelm Plesken, Gabriele Nebe, and Bernd Souvignier for supplying their results to GAP.

In the preceding acknowledgement, we used some notations that will also be needed in the sequel. We first define these.

Any integral matrix group G of dimension n is a subgroup of GL_n(Z) as well as of GL_n(Q) and hence lies in some conjugacy class of integral matrix groups under GL_n(Z) and also in some conjugacy class of rational matrix groups under GL_n(Q). As usual, we call these classes the Z-class and the Q-class of G, respectively. Note that any conjugacy class of subgroups of GL_n(Q) contains at least one Z-class of subgroups of GL_n(Z) and hence can be considered as the Q-class of some integral matrix group.

In the context of this library we are only concerned with Z-classes and Q-classes of subgroups of GL_n(Z) which are irreducible and maximal finite in GL_n(Z) (we will call them em i.,m.,f.~subgroups of GL_n(Z)). We can distinguish two types of these groups:

First, there are those i.,m.,f.~subgroups of GL_n(Z) which are also maximal finite subgroups of GL_n(Q). Let us denote the set of their Q-classes by Q_1(n). It is clear from the above remark that Q_1(n) just consists of the Q-classes of, i.,m.,f.~subgroups of GL_n(Q).

Secondly, there is the set Q_2(n) of the Q-classes of the remaining i.,m.,f.~subgroups of GL_n(Z), i.,e., of those which are not maximal finite subgroups of GL_n(Q). For any such group G, say, there is at least one class C in Q_1(n) such that G is conjugate under Q to a proper subgroup of some group H in C. In fact, the class C is uniquely determined for any group G occurring in the library (though there seems to be no reason to assume that this property should hold in general). Hence we may call C the em rational i.,m.,f.~class of G. Finally, we will denote the number of classes in Q_1(n) and Q_2(n) by q_1(n) and q_2(n), respectively.

As an example, let us consider the case n = 4. There are 6 Z-classes of, i.,m.,f.~subgroups of GL_4(Z) with representative subgroups G_1, ldots, G_6 of isomorphsim types, mboxG_1 cong W(F_4),, mboxG_2 cong D_{12} wr C_2,, mboxG_3 cong G_4 cong C_2 times S_5,, mboxG_5 cong W(B_4),, and, mboxG_6 cong (D_{12} {sf Y} D_{12}) !:! C_2. The corresponding Q-classes, R_1, ldots, R_6, say, are pairwise different except that R_3 coincides with R_4. The groups G_1, G_2, and G_3 are i.,m.,f.~subgroups of GL_4(Q), but G_5 and G_6 are not because they are conjugate under GL_4(Q) to proper subgroups of G_1 and G_2, respectively. So we have Q_1(4) = { R_1, R_2, R_3 },, Q_2(4) = { R_5, R_6 },, q_1(4) = 3, and q_2(4) = 2.

The q_1(n) Q-classes of, i.,m.,f.~subgroups of GL_n(Q) have been determined for each dimension n leq 24. The current GAP library provides integral representative groups for all these classes. Moreover, all Z-classes of, i.,m.,f.~subgroups of GL_n(Z) are known for n leq 11 and for n in {13,17,19,23}. For these dimensions, the library offers integral representative groups for all Q-classes in Q_1(n) and Q_2(n) as well as for all Z-classes of, i.,m.,f.~subgroups of GL_n(Z).

Any group G of dimension n given in the library is represented as the automorphism group G = {rm Aut}(F,L) = { g in GL_n(Z) mid Lg = L ; {rm and} ; g F g^{rm tr} = F } of a positive definite symmetric n times n matrix F in Z^{n times n} on an n-dimensional lattice L cong Z^{1 times n} (for details see e.,g. PN95). GAP provides for G a list of matrix generators and the em Gram matrix F.

The positive definite quadratic form defined by F defines a em norm v F v^{rm tr} for each vector v in L, and there is only a finite set of vectors of minimal norm. These vectors are often simply called the em ``short vectors''. Their set splits into orbits under G, and G being irreducible acts faithfully on each of these orbits by multiplication from the right. GAP provides for each of these orbits the orbit size and a representative vector.

Like most of the other GAP libraries, the library of, i.,m.,f.~integral matrix groups supplies an extraction function, `ImfMatGroup`. However, as the library involves only 423 different groups, there is no need for a selection or an example function. Instead, there are two functions, `ImfInvariants` and `DisplayImfInvariants`, which provide some Z-class invariants that can be extracted from the library without actually constructing the representative groups themselves. The difference between these two functions is that the latter one displays the resulting data in some easily readable format, whereas the first one returns them as record components so that you can properly access them.

We shall give an individual description of each of the library functions, but first we would like to insert a short remark concerning their names: Any self-explaining name of a function handling em irreducible maximal finite integral matrix groups would have to include this term in full length and hence would grow extremely long. Therefore we have decided to use the abbreviation `Imf` instead in order to restrict the names to some reasonable length.

The first three functions can be used to formulate loops over the classes.

`ImfNumberQQClasses( dim )` `ImfNumberQClasses( dim )` `ImfNumberZClasses( dim, q )`

`ImfNumberQQClasses` returns the number q_1(dim) of Q-classes of, i.,m.,f.~rational matrix groups of dimension dim. Valid values of dim are all positive integers up to 24.

Note: In order to enable you to loop just over the classes belonging to Q_1(dim), we have arranged the list of Q-classes of dimension dim for any dimension dim in the library such that, whenever the classes of Q_2(dim) are known, too, i.,e., in the cases dim leq 11 or dim in {13,17,19,23}, the classes of Q_1(dim) precede those of Q_2(dim) and hence are numbered from 1 to q_1(dim).

`ImfNumberQClasses` returns the number of Q-classes of groups of dimension dim which are available in the library. If dim leq 11 or dim in {13,17,19,23}, this is the number q_1(dim) + q_2(dim) of Q-classes of, i.,m.,f.~subgroups of GL_{dim}(Z). Otherwise, it is just the number q_1(dim) of Q-classes of, i.,m.,f.~subgroups of GL_{dim}(Q). Valid values of dim are all positive integers up to 24.

`ImfNumberZClasses` returns the number of Z-classes in the q^{rm th} Q-class of, i.,m.,f.~integral matrix groups of dimension dim. Valid values of dim are all positive integers up to 11 and all primes up to 23.

`DisplayImfInvariants( dim, q )` `DisplayImfInvariants( dim, q, z )`

`DisplayImfInvariants` displays the following Z-class invariants of the groups in the z^{rm th} Z-class in the q^{rm th} Q-class of i.,m.,f.~integral matrix groups of dimension dim: item its Z-class number in the form dim.q.z, if dim is at most 11 or a prime, or its Q-class number in the form dim.q, else, vspace-2mm item a message if the group is solvable, vspace-2mm item the size of the group, vspace-2mm item the isomorphism type of the group, vspace-2mm item the elementary divisors of the associated quadratic form, vspace-2mm item the sizes of the orbits of short vectors (these sizes are the degrees of the faithful permutation representations which you may construct using the `PermGroup` or `PermGroupImfGroup` commands below), vspace-2mm item the norm of the associated short vectors, vspace-2mm item only in case that the group is not an i.,m.,f.~group in GL_n(Q): an appropriate message, including the Q-class number of the corresponding rational i.,m.,f.~class. vspace-2mm If you specify the value 0 for any of the parameters dim, q, or z, the command will loop over all available dimensions, Q-classes of given dimension, or Z-classes within the given Q-class, respectively. Otherwise, the values of the arguments must be in range. A value z neq 1 must not be specified if the Z-classes are not known for the given dimension, i.,e., if dim > 11 and dim not in {13,17,19,23}. The default value of z is~1. This value of z will be accepted even if the Z-classes are not known. Then it specifies the only representative group which is available for the q^{rm th} Q-class. The greatest legal value of dim is 24.

```    gap> DisplayImfInvariants( 3, 1, 0 );
#I Z-class 3.1.1:  Solvable, size = 2^4*3
#I   isomorphism type = C2 wr S3 = C2 x S4 = W(B3)
#I   elementary divisors = 1^3
#I   orbit size = 6, minimal norm = 1
#I Z-class 3.1.2:  Solvable, size = 2^4*3
#I   isomorphism type = C2 wr S3 = C2 x S4 = C2 x W(A3)
#I   elementary divisors = 1*4^2
#I   orbit size = 8, minimal norm = 3
#I Z-class 3.1.3:  Solvable, size = 2^4*3
#I   isomorphism type = C2 wr S3 = C2 x S4 = C2 x W(A3)
#I   elementary divisors = 1^2*4
#I   orbit size = 12, minimal norm = 2
gap> DisplayImfInvariants( 8, 15, 1 );
#I Z-class 8.15.1:  Solvable, size = 2^5*3^4
#I   isomorphism type = C2 x (S3 wr S3)
#I   elementary divisors = 1*3^3*9^3*27
#I   orbit size = 54, minimal norm = 8
#I   not maximal finite in GL(8,Q), rational imf class is 8.5
gap> DisplayImfInvariants( 20, 23 );
#I Q-class 20.23:  Size = 2^5*3^2*5*11
#I   isomorphism type = (PSL(2,11) x D12).C2
#I   elementary divisors = 1^18*11^2
#I   orbit size = 3*660 + 2*1980 + 2640 + 3960, minimal norm = 4 ```

Note that the `DisplayImfInvariants` function uses a kind of shorthand to display the elementary divisors. E.~g., the expression `1*3^3*9^3*27` in the preceding example stands for the elementary divisors 1,3,3,3,9,9,9,27. (See also the next example which shows that the `ImfInvariants` function provides the elementary divisors in form of an ordinary GAP list.)

In the description of the isomorphism types the following notations are used: item[] item[] item[makebox[15mm][l]A,mbox{x},B] denotes a direct product of a group A by a group B, vspace-1mm item[makebox[15mm][l]A,mbox{subd},B] denotes a subdirect product of A by B, vspace-1mm item[makebox[15mm][l]A,mbox{Y},B] denotes a central product of A by B, vspace-1mm item[makebox[15mm][l]A,mbox{wr},B] denotes a wreath product of A by B, vspace-1mm item[makebox[15mm][l]A:B] denotes a split extension of A by B, vspace-1mm item[makebox[15mm][l]A,.,B] denotes just an extension of A by B (split or nonsplit). vspace-2mm The groups involved are item the cyclic groups C_n, dihedral groups D_n, and generalized quaternion groups Q_n of order n, denoted by `C`n, `D`n, and `Q`n, respectively, vspace-2mm item the alternating groups A_n and symmetric groups S_n of degree n, denoted by `A`n and `S`n, respectively, vspace-2mm item the linear groups GL_n(q), PGL_n(q), SL_n(q), and PSL_n(q), denoted by `GL(`n`,`q`)`, `PGL(`n`,`q`)`, `SL(`n`,`q`)`, and `PSL(`n`,`q`)`, respectively, vspace-2mm item the unitary groups SU_n(q) and PSU_n(q), denoted by `SU(`n`,`q`)` and `PSU(`n`,`q`)`, respectively, vspace-2mm item the symplectic groups Sp(n,q), denoted by `Sp(`n`,`q`)`, vspace-2mm item the orthogonal group O_8^{,+}(2), denoted by `O+(8,2)`, vspace-2mm item the extraspecial groups 2_+^{,1+8}, 3_+^{,1+2}, 3_+^{,1+4}, and 5_+^{,1+2}, denoted by `2+^(1+8)`, `3+^(1+2)`, `3+^(1+4)`, and `5+^(1+2)`, respectively, vspace-2mm item the Chevalley group G_2(3), denoted by `G(2,3)`, vspace-2mm item the Weyl groups W(A_n), W(B_n), W(D_n), W(E_n), and W(F_4), denoted by `W(A`n`)`, `W(B`n`)`, `W(D`n`)`, `W(E`n`)`, and `W(F4)`, respectively, vspace-2mm item the sporadic simple groups Co_1, Co_2, Co_3, HS, J_2, M_{12}, M_{22}, M_{23}, M_{24}, and Mc, denoted by `Co1`, `Co2`, `Co3`, `HS`, `J2`, `M12`, `M22`, `M23`, `M24`, and `Mc`, respectively, vspace-2mm item a point stabilizer of index 11 in M_{11}, denoted by `M10`. vspace-2mm

As mentioned above, the data assembled by the `DisplayImfInvariants` command are ``cheap data'' in the sense that they can be provided by the library without loading any of its large matrix files or performing any matrix calculations. The following function allows you to get proper access to these cheap data instead of just displaying them.

`ImfInvariants( dim, q )` `ImfInvariants( dim, q, z )`

`ImfInvariants` returns a record which provides some Z-class invariants of the groups in the z^{rm th} Z-class in the q^{rm th} Q-class of i.,m.,f.~integral matrix groups of dimension dim. A value z neq 1 must not be specified if the Z-classes are not known for the given dimension, i.,e., if dim > 11 and dim not in {13,17,19,23}. The default value of z is~1. This value of z will be accepted even if the Z-classes are not known. Then it specifies the only representative group which is available for the q^{rm th} Q-class. The greatest legal value of dim is 24.

The resulting record contains six or seven components:

`size`:

the size of any representative group G,

`isSolvable`:

is `true` if G is solvable,

`isomorphismType`:

a text string describing the isomorphism type of G (in the same notation as used by the `DisplayImfInvariants` command above),

`elementaryDivisors`:

the elementary divisors of the associated Gram matrix F (in the same format as the result of the `ElementaryDivisorsMat` function, see ElementaryDivisorsMat),

`minimalNorm`:

the norm of the associated short vectors,

`sizesOrbitsShortVectors`:

the sizes of the orbits of short vectors under F,

`maximalQClass`:

the Q-class number of an i.,m.,f.~group in GL_n(Q) that contains G as a subgroup (only in case that not G itself is an i.,m.,f.~subgroup of GL_n(Q)).

Note that four of these data, namely the group size, the solvability, the isomorphism type, and the corresponding rational i.,m.,f.~class, are not only Z-class invariants, but also Q-class invariants.

Note further that, though the isomorphism type is a Q-class invariant, you will sometimes get different descriptions for different Z-classes of the same Q-class (as, e.,g., for the classes 3.1.1 and 3.1.2 in the last example above). The purpose of this behaviour is to provide some more information about the underlying lattices.

```    gap> ImfInvariants( 8, 15, 1 );
rec(
size := 2592,
isSolvable := true,
isomorphismType := "C2 x (S3 wr S3)",
elementaryDivisors := [ 1, 3, 3, 3, 9, 9, 9, 27 ],
minimalNorm := 8,
sizesOrbitsShortVectors := [ 54 ],
maximalQClass := 5 )
gap> ImfInvariants( 24, 1 ).size;
10409396852733332453861621760000
gap> ImfInvariants( 23, 5, 2 ).sizesOrbitsShortVectors;
[ 552, 53130 ]
gap> for i in [ 1 .. ImfNumberQClasses( 22 ) ] do
>    Print( ImfInvariants( 22, i ).isomorphismType, "\n" ); od;
C2 wr S22 = W(B22)
(C2 x PSU(6,2)).S3
(C2 x S3) wr S11 = (C2 x W(A2)) wr S11
(C2 x S12) wr C2 = (C2 x W(A11)) wr C2
C2 x S3 x S12 = C2 x W(A2) x W(A11)
(C2 x HS).C2
(C2 x Mc).C2
C2 x S23 = C2 x W(A22)
C2 x PSL(2,23)
C2 x PSL(2,23)
C2 x PGL(2,23)
C2 x PGL(2,23) ```

`ImfMatGroup( dim, q )` `ImfMatGroup( dim, q, z )`

`ImfMatGroup` is the essential extraction function of this library. It returns a representative group, G say, of the z^{rm th} Z-class in the q^{rm th} Q-class of i.,m.,f.~integral matrix groups of dimension dim. A value z neq 1 must not be specified if the Z-classes are not known for the given dimension, i.,e., if dim > 11 and dim not in {13,17,19,23}. The default value of z is~1. This value of z will be accepted even if the Z-classes are not known. Then it specifies the only representative group which is available for the q^{rm th} Q-class. The greatest legal value of dim is 24.

```    gap> G := ImfMatGroup( 5, 1, 3 );
ImfMatGroup(5,1,3)
gap> for m in G.generators do PrintArray( m ); od;
[ [  -1,   0,   0,   0,   0 ],
[   0,   1,   0,   0,   0 ],
[   0,   0,   0,   1,   0 ],
[  -1,  -1,  -1,  -1,   2 ],
[  -1,   0,   0,   0,   1 ] ]
[ [  0,  1,  0,  0,  0 ],
[  0,  0,  1,  0,  0 ],
[  0,  0,  0,  1,  0 ],
[  1,  0,  0,  0,  0 ],
[  0,  0,  0,  0,  1 ] ] ```

The group record of G contains the usual components of a matrix group record. In addition, it includes the same six or seven records as the resulting record of the `ImfInvariants` function described above, namely the components `size`, `isSolvable`, `isomorphismType`, `elementaryDivisors`, `minimalNorm`, and `sizesOrbitsShortVectors` and, if G is not a rational i.,m.,f.~group, `maximalQClass`. Moreover, there are the two components

`form`:

the associated Gram matrix F,

`repsOrbitsShortVectors`:

representatives of the orbits of short vectors under F.

The last of these components will be required by the `PermGroup` function below.

Example:

```    gap> G.size;
3840
gap> G.isomorphismType;
"C2 wr S5 = C2 x W(D5)"
gap> PrintArray( G.form );
[ [  4,  0,  0,  0,  2 ],
[  0,  4,  0,  0,  2 ],
[  0,  0,  4,  0,  2 ],
[  0,  0,  0,  4,  2 ],
[  2,  2,  2,  2,  5 ] ]
gap> G.elementaryDivisors;
[ 1, 4, 4, 4, 4 ]
gap> G.minimalNorm;
4 ```

If you want to perform calculations in such a matrix group G you should be aware of the fact that GAP offers much more efficient permutation group routines than matrix group routines. Hence we recommend that you do your computations, whenever it is possible, in the isomorphic permutation group that is induced by the action of G on one of the orbits of the associated short vectors. You may call one of the following functions to get such a permutation group.

`PermGroup( G )`

`PermGroup` returns the permutation group which is induced by the given i.,m.,f.~integral matrix group G on an orbit of minimal size of G on the set of short vectors (see also `PermGroupImfGroup` below).

The permutation representation is computed by first constructing the specified orbit, S say, of short vectors and then computing the permutations which are induced on S by the generators of G. We would like to warn you that in case of a large orbit this procedure may be space and time consuming. Fortunately, there are only five Q-classes in the library for which the smallest orbit of short vectors is of size greater than 20000, the worst case being the orbit of size 196560 for the Leech lattice (dim = 24, q = 3).

As mentioned above, `PermGroup` constructs the required permutation group, P say, as the image of G under the isomorphism between the matrices in G and their action on S. Moreover, it constructs the inverse isomorphism from P to G, varphi say, and returns it in the group record component P.`bijection` of P. In fact, varphi is constructed by determining a Q-base B subset S of Q^{1 times dim} in S and, in addition, the associated base change matrix M which transforms B into the standard base of Z^{1 times dim}. Then the image varphi(p) of any permutation p in P can be easily computed: If, for 1 leq i leq dim, b_i is the position number in S of the i^{rm th} base vector in B, it suffices to look up the vector whose position number in S is the image of b_i under p and to multiply this vector by M to get the i^{rm th} row of varphi(p).

You may use varphi at any time to compute the images in G of permutations in P or to compute the preimages in P of matrices in G.

The record of P contains, in addition to the usual components of permutation group records, some special components. The most important of those are:

`isomorphismType`:

a text string describing the isomorphism type of P (in the same notation as used by the `DisplayImfInvariants` command above),

`matGroup`:

the associated matrix group G,

`bijection`:

the isomorphism varphi from P to G,

`orbitShortVectors`:

the orbit S of short vectors (needed for varphi),

`baseVectorPositions`:

the position numbers of the base vectors in B with respect to S (needed for varphi),

`baseChangeMatrix`:

the base change matrix M (needed for varphi).

As an example, let us compute a set R of matrix generators for the solvable residuum of the group G that we have constructed in the preceding example.

```    gap> # Perform the computations in an isomorphic permutation group.
gap> P := PermGroup( G );
PermGroup(ImfMatGroup(5,1,3))
gap> P.generators;
[ ( 1, 7, 6)( 2, 9)( 4, 5,10), ( 2, 3, 4, 5)( 6, 9, 8, 7) ]
gap> D := DerivedSubgroup( P );
Subgroup( PermGroup(ImfMatGroup(5,1,3)),
[ ( 1, 2,10, 9)( 3, 8)( 4, 5)( 6, 7),
( 1, 6)( 2, 7, 9, 4)( 3, 8)( 5,10), ( 1, 5,10, 6)( 2, 8, 9, 3) ] )
gap> Size( D );
960
gap> IsPerfect( D );
true
gap> # Now move the results back to the matrix group.
gap> phi := P.bijection;;
gap> R := List( D.generators, p -> Image( phi, p ) );;
gap> for m in R do PrintArray( m ); od;
[ [  -1,  -1,  -1,  -1,   2 ],
[   0,  -1,   0,   0,   0 ],
[   0,   0,   0,   1,   0 ],
[   0,   0,   1,   0,   0 ],
[  -1,  -1,   0,   0,   1 ] ]
[ [   0,   0,  -1,   0,   0 ],
[   0,  -1,   0,   0,   0 ],
[   1,   0,   0,   0,   0 ],
[  -1,  -1,  -1,  -1,   2 ],
[   0,  -1,  -1,   0,   1 ] ]
[ [   0,  -1,   0,   0,   0 ],
[   1,   0,   0,   0,   0 ],
[   0,   0,   1,   0,   0 ],
[  -1,  -1,  -1,  -1,   2 ],
[   0,  -1,   0,  -1,   1 ] ]
gap> # The PreImage function allows us to use the inverse of phi.
gap> PreImage( phi, R[1] ) = D.generators[1];
true ```

`PermGroupImfGroup( G, n )`

`PermGroupImfGroup` returns the permutation group which is induced by the given i.,m.,f.~integral matrix group G on its n^{rm th} orbit of short vectors. The only difference to the above `PermGroup` function is that you can specify the orbit to be used. In fact, as the orbits of short vectors are sorted by increasing sizes, the function ```PermGroup( G )``` has been implemented such that it is equivalent to `PermGroupImfGroup( G, 1 )`.

```    gap> ImfInvariants( 12, 9 ).sizesOrbitsShortVectors;
[ 120, 300 ]
gap> G := ImfMatGroup( 12, 9 );
ImfMatGroup(12,9)
gap> P1 := PermGroupImfGroup( G, 1 );
PermGroup(ImfMatGroup(12,9))
gap> P1.degree;
120
gap> P2 := PermGroupImfGroup( G, 2 );
PermGroupImfGroup(ImfMatGroup(12,9),2)
gap> P2.degree;
300 ```

## 37.13 The Crystallographic Groups Library

GAP provides a library of crystallographic groups of dimensions 2, 3, and 4 which covers many of the data that have been listed in the book ``Crystallographic groups of four-dimensional space'' BBNWZ78. It has been brought into GAP format by Volkmar Felsch.

How to access the data of the book

Among others, the library offers functions which provide access to the data listed in the Tables 1, 5, and 6 of BBNWZ78: item The information on the crystal families listed in Table 1 can be reproduced using the `DisplayCrystalFamily` function. item Similarly, the `DisplayCrystalSystem` function can be used to reproduce the information on the crystal systems provided in Table 1. item The information given in the Q-class headlines of Table 1 can be displayed by the `DisplayQClass` function, whereas the `FpGroupQClass` function can be used to reproduce the presentations that are listed in Table 1 for the Q-class representatives. item The information given in the Z-class headlines of Table 1 will be covered by the results of the `DisplayZClass` function, and the matrix generators of the Z-class representatives can be constructed by calling the `MatGroupZClass` function. item The `DisplaySpaceGroupType` and the `DisplaySpaceGroupGenerators` functions can be used to reproduce all of the information on the space-group types that is provided in Table 1. item The normalizers listed in Table 5 can be reproduced by calling the `NormalizerZClass` function. item Finally, the `CharTableQClass` function will compute the character tables listed in Table 6, whereas the isomorphism types given in Table 6 may be obtained by calling the `DisplayQClass` function. vspace-2mm The display functions mentioned in the above list print their output with different indentation. So, calling them in a suitably nested loop, you may produce a listing in which the information about the objects of different type will be properly indented as has been done in Table 1 of BBNWZ78.

Representation of space groups in GAP

Probably the most important function in the library is the `SpaceGroup` function which provides representatives of the affine classes of space groups. A space group of dimension n is represented by an (n+1)-dimensional rational matrix group as follows.

If S is an n-dimensional space group, then each element alpha in S is an affine mapping alpha!: V rightarrow V of an n-dimensional R-vector space V onto itself. Hence alpha can be written as the sum of an appropriate invertible linear mapping varphi!: V rightarrow V and a translation by some translation vector t in V such that, if we write mappings from the left, we have alpha(v) = varphi(v) + t for all v in V.

If we fix a basis of V and then replace each v in V by the column vector of its coefficients with respect to that basis (and hence V by the isomorphic column vector space R^{n times 1}), we can describe the linear mapping varphi involved in alpha by an n times n matrix M_varphi in GL_n(R) which acts by multiplication from the left on the column vectors in R^{n times 1}. Hence, if we identify V with R^{n times 1}, we have alpha(v) = M_varphi cdot v + t for all v in R^{n times 1}.

Moreover, if we extend each column vector v in R^{n times 1} to a column left[ ! ! catcode`|=12 begin{tabular}{c} catcode`|=13 v \ 1 end{tabular} ! ! right] of length n+1 by adding an entry 1 in the last position and if we define an (n+1) times (n+1) matrix M_alpha = left[ catcode`|=12 begin{tabular}{c|c} catcode`|=13 M_varphi & t \ hline 0 & 1 end{tabular} right], we have left[ ! ! catcode`|=12 begin{tabular}{c} catcode`|=13 alpha(v) \ 1 for all v in R^n times 1. This means that we can represent the space group S by the isomorphic group M(S) = { M_alpha mid alpha in S }. The submatrices M_varphi occurring in the elements of M(S) form an n times n matrix group P(S), the ``point group'' of M(S). In fact, we can choose the basis of R^n times 1 such that M_varphi in GL_n(Z) and t in Q^n times 1 for all M_alpha in M(S). In particular, the space group representatives that are normally used by the crystallographers are of this form, and the book BBNWZ78 uses the same convention.

Before we describe all available library functions in detail, we have to add three remarks.

Remark 1

The concepts used in this section are defined in chapter 1 (Basic definitions) of BBNWZ78. However, note that the definition of the concept of a crystal system given on page 16 of that book relies on the following statement about Q-classes: item[] For a Q-class C there is a unique holohedry H such that each f.,u. group in C is a subgroup of some f.,u. group in H, but is not a subgroup of any f.,u. group belonging to a holohedry of smaller order. vspace-2mm This statement is correct for dimensions 1, 2, 3, and 4, and hence the definition of ``crystal system'' given on page 16 of BBNWZ78 is known to be unambiguous for these dimensions. However, there is a counterexample to this statement in seven-dimensional space so that the definition breaks down for some higher dimensions.

Therefore, the authors of the book have since proposed to replace this definition of ``crystal system'' by the following much simpler one, which has been discussed in more detail in NPW81. To formulate it, we use the intersections of Q-classes and Bravais flocks as introduced on page 17 of BBNWZ78, and we define the classification of the set of all Z-classes into crystal systems as follows: item[] Definition: A crystal system (introduced as an equivalence class of Z-classes) consists of full geometric crystal classes. The Z-classes of two (geometric) crystal classes belong to the same crystal system if and only if these geometric crystal classes intersect the same set of Bravais flocks of Z-classes. vspace-2mm From this definition of a crystal system of Z-classes one then obtains crystal systems of f.,u. groups, of space-group types, and of space groups in the same manner as with the preceding definitions in the book.

The new definition is unambiguous for all dimensions. Moreover, it can be checked from the tables in the book that it defines the same classification as the old one for dimensions 1, 2, 3, and 4.

It should be noted that the concept of crystal family is well-defined independently of the dimension if one uses the ``more natural'' second definition of it at the end of page 17. Moreover, the first definition of crystal family on page 17 defines the same concept as the second one if the now proposed definition of crystal system is used.

Remark 2

The second remark just concerns a different terminology in the tables of BBNWZ78 and in the current library. In group theory, the number of elements of a finite group usually is called the ``order'' of the group. This notation has been used throughout in the book. Here, however, we will follow the GAP conventions and use the term ``size'' instead.

Remark 3

The third remark concerns a problem in the use of the space groups that should be well understood.

There is an alternative to the representation of the space group elements by matrices of the form left[ catcode`|=12 begin{tabular}{c|c} catcode`|=13 M_varphi & t \ hline 0 & 1 end{tabular} right] as described above. Instead of considering the coefficient vectors as columns we may consider them as rows. Then we can associate to each affine mapping alpha in S an (n+1) times (n+1) matrix overline{M}_alpha = left[ catcode`|=12 begin{tabular}{c|c} catcode`|=13 overlineM_overlinevarphi & 0 \ hline overlinet & 1 end{tabular} right] with overline{M}_{overline{varphi}} in GL_n(R) and overline{t} in R^{1 times n} such that [alpha(overline{v}),1] = [overline{v},1] cdot overline{M}_alpha for all overline{v} in R^{1 times n}, and we may represent S by the matrix group overline{M}(S) = { overline{M}_alpha mid alpha in S }. Again, we can choose the basis of R^{1 times n} such that overline{M}_{overline{varphi}} in GL_n(Z) and overline{t} in Q^{1 times n} for all overline{M}_alpha in overline{M}(S).

From the mathematical point of view, both approaches are equivalent. In particular, M(S) and overline{M}(S) are isomorphic, for instance via the isomorphism tau mapping M_alpha in M(S) to (M_alpha^{rm tr})^{-1}. Unfortunately, however, neither of the two is a good choice for our GAP library.

The first convention, using matrices which act on column vectors from the left, is not consistent with the fact that actions in GAP are usually from the right.

On the other hand, if we choose the second convention, we run into a problem with the names of the space groups as introduced in BBNWZ78. Any such name does not just describe the abstract isomorphism type of the respective space group S, but reflects properties of the matrix group M(S). In particular, it contains as a leading part the name of the Z-class of the associated point group P(S). Since the classification of space groups by affine equivalence is tantamount to their classification by abstract isomorphism, overline{M}(S) lies in the same affine class as M(S) and hence should get the same name as M(S). But the point group P(S) that occurs in that name is not always Z-equivalent to the point group overline{P}(S) of overline{M}(S). For example, the isomorphism tau!!: M(S) rightarrow overline{M}(S) defined above maps the Z-class representative with the parameters [3,7,3,2] (in the notation described below) to the Z-class representative with the parameters [3,7,3,3]. In other words: The space group names introduced for the groups M(S) in BBNWZ78 lead to confusing inconsistencies if assigned to the groups overline{M}(S).

In order to avoid this confusion we decided that the first convention is the lesser evil. So the GAP library follows the book, and if you call the `SpaceGroup` function you will get the same space group representatives as given there. This does not cause any problems as long as you do calculations within these groups treating them just as matrix groups of certain isomorphism types. However, if it is necesary to consider the action of a space group as affine mappings on the natural lattice, you need to use the transposed representation of the space group. For this purpose the library offers the `TransposedSpaceGroup` function which not only transposes the matrices, but also adapts appropriately the associated group presentation.

Both these functions are described in detail in the following.

The library functions

`NrCrystalFamilies( dim )`

`NrCrystalFamilies` returns the number of crystal families in case of dimension dim. It can be used to formulate loops over the crystal families.

There are 4, 6, and 23 crystal families of dimension 2, 3, and 4, respectively.

```    gap> n := NrCrystalFamilies( 4 );
23 ```

`DisplayCrystalFamily( dim, family )`

`DisplayCrystalFamily` displays for the specified crystal family essentially the same information as is provided for that family in Table 1 of BBNWZ78, namely item the family name, vspace-2mm item the number of parameters, vspace-2mm item the common rational decomposition pattern, vspace-2mm item the common real decomposition pattern, vspace-2mm item the number of crystal systems in the family, and vspace-2mm item the number of Bravais flocks in the family. vspace-2mm For details see BBNWZ78.

```    gap> DisplayCrystalFamily( 4, 17 );
#I Family XVII: cubic orthogonal; 2 free parameters;
#I  Q-decomposition pattern 1+3; R-decomposition pattern 1+3;
#I  2 crystal systems; 6 Bravais flocks
gap> DisplayCrystalFamily( 4, 18 );
#I Family XVIII: octagonal; 2 free parameters;
#I  Q-irreducible; R-decomposition pattern 2+2;
#I  1 crystal system; 1 Bravais flock
gap> DisplayCrystalFamily( 4, 21 );
#I Family XXI: di-isohexagonal orthogonal; 1 free parameter;
#I  R-irreducible; 2 crystal systems; 2 Bravais flocks ```

`NrCrystalSystems( dim )`

`NrCrystalSystems` returns the number of crystal systems in case of dimension dim. It can be used to formulate loops over the crystal systems.

There are 4, 7, and 33 crystal systems of dimension 2, 3, and 4, respectively.

```    gap> n := NrCrystalSystems( 2 );
4 ```

The following two functions are functions of crystal systems.

Each crystal system is characterized by a pair `(dim,,system)` where dim is the associated dimension, and system is the number of the crystal system.

`DisplayCrystalSystem( dim, system )`

`DisplayCrystalSystem` displays for the specified crystal system essentially the same information as is provided for that system in Table 1 of BBNWZ78, namely item the number of Q-classes in the crystal system and vspace-2mm item the identification number, i.,e., the tripel `(dim,,system,,q-class)` described below, of the Q-class that is the holohedry of the crystal system. vspace-2mm For details see BBNWZ78.

```    gap> for sys in [ 1 .. 4 ] do  DisplayCrystalSystem( 2, sys );  od;
#I  Crystal system 1: 2 Q-classes; holohedry (2,1,2)
#I  Crystal system 2: 2 Q-classes; holohedry (2,2,2)
#I  Crystal system 3: 2 Q-classes; holohedry (2,3,2)
#I  Crystal system 4: 4 Q-classes; holohedry (2,4,4) ```

`NrQClassesCrystalSystem( dim, system )`

`NrQClassesCrystalSystem` returns the number of Q-classes within the given crystal system. It can be used to formulate loops over the Q-classes.

The following five functions are functions of Q-classes.

In general, the parameters characterizing a Q-class will form a triple `(dim,,system,,q-class)` where dim is the associated dimension, system is the number of the associated crystal system, and q-class is the number of the Q-class within the crystal system. However, in case of dimensions 2 or 3, a Q-class may also be characterized by a pair `(dim, IT-number)` where IT-number is the number in the International Tables for Crystallography Hah83 of any space-group type lying in (a Z-class of) that Q-class, or just by the Hermann-Mauguin symbol of any space-group type lying in (a Z-class of) that Q-class.

The Hermann-Mauguin symbols indexHermann-Mauguin symbol which we use in GAP are the short Hermann-Mauguin symbols defined in the 1983 edition of the International Tables Hah83, but any occurring indices are expressed by ordinary integers, and bars are replaced by minus signs. For example, the Hermann-Mauguin symbol Poverline{4}2_1m will be represented by the string `"P-421m"`.

`DisplayQClass( dim, system, q-class )` `DisplayQClass( dim, IT-number )`
`DisplayQClass( Hermann-Mauguin-symbol )`

`DisplayQClass` displays for the specified Q-class essentially the same information as is provided for that Q-class in Table 1 of BBNWZ78 (except for the defining relations given there), namely item the size of the groups in the Q-class, vspace-2mm item the isomorphism type of the groups in the Q-class, vspace-2mm item the Hurley pattern, vspace-2mm item the rational constituents, vspace-2mm item the number of Z-classes in the Q-class, and vspace-2mm item the number of space-group types in the Q-class. vspace-2mm For details see BBNWZ78.

```    gap> DisplayQClass( "p2" );
#I   Q-class H (2,1,2): size 2; isomorphism type 2.1 = C2;
#I    Q-constituents 2*(2,1,2); cc; 1 Z-class; 1 space group
gap> DisplayQClass( "R-3" );
#I   Q-class (3,5,2): size 6; isomorphism type 6.1 = C6;
#I    Q-constituents (3,1,2)+(3,4,3); ncc; 2 Z-classes; 2 space grps
gap> DisplayQClass( 3, 195 );
#I   Q-class (3,7,1): size 12; isomorphism type 12.5 = A4;
#I    C-irreducible; 3 Z-classes; 5 space grps
gap> DisplayQClass( 4, 27, 4 );
#I   Q-class H (4,27,4): size 20; isomorphism type 20.3 = D10xC2;
#I    Q-irreducible; 1 Z-class; 1 space group
gap> DisplayQClass( 4, 29, 1 );
#I  *Q-class (4,29,1): size 18; isomorphism type 18.3 = D6xC3;
#I    R-irreducible; 3 Z-classes; 5 space grps ```

Note in the preceding examples that, as pointed out above, the term ``size'' denotes the order of a representative group of the specified Q-class and, of course, not the (infinite) class length.

`FpGroupQClass( dim, system, q-class )` `FpGroupQClass( dim, IT-number )`
`FpGroupQClass( Hermann-Mauguin-symbol )`

`FpGroupQClass` returns a finitely presented group F, say, which is isomorphic to the groups in the specified Q-class.

The presentation of that group is the same as the corresponding presentation given in Table 1 of BBNWZ78 except for the fact that its generators are listed in reverse order. The reason for this change is that, whenever the group in question is solvable, the resulting generators form an AG system (as defined in GAP) if they are numbered ``from the top to the bottom'', and the presentation is a polycyclic power commutator presentation. The `AgGroupQClass` function described next will make use of this fact in order to construct an ag group isomorphic to F.

Note that, for any Z-class in the specified Q-class, the matrix group returned by the `MatGroupZClass` function (see below) not only is isomorphic to F, but also its generators satisfy the defining relators of F.

Besides of the usual components, the group record of F will have an additional component F`.crQClass` which saves a list of the parameters that specify the given Q-class.

```    gap> F := FpGroupQClass( 4, 20, 3 );
FpGroupQClass( 4, 20, 3 )
gap> F.generators;
[ f.1, f.2 ]
gap> F.relators;
[ f.1^2*f.2^-3, f.2^6, f.2^-1*f.1^-1*f.2*f.1*f.2^-4 ]
gap> F.size;
12
gap> F.crQClass;
[ 4, 20, 3 ] ```

`AgGroupQClass( dim, system, q-class )` `AgGroupQClass( dim, IT-number )`
`AgGroupQClass( Hermann-Mauguin-symbol )`

`AgGroupQClass` returns an ag group A, say, isomorphic to the groups in the specified Q-class, if these groups are solvable, or the value `false` (together with an appropriate warning), otherwise.

A is constructed by first establishing a finitely presented group (as it would be returned by the `FpGroupQClass` function described above) and then constructing from it an isomorphic ag group. If the underlying AG system is not yet a PAG system (see sections More about Ag Words and More about Ag Groups), it will be refined appropriately (and a warning will be displayed).

Besides of the usual components, the group record of A will have an additional component A`.crQClass` which saves a list of the parameters that specify the given Q-class.

```    gap> A := AgGroupQClass( 4, 31, 3 );
#I  Warning: a non-solvable group can't be represented as an ag group
false
gap> A := AgGroupQClass( 4, 20, 3 );
#I  Warning: the presentation has been extended to get a PAG system
AgGroupQClass( 4, 20, 3 )
gap> A.generators;
[ f.1, f.21, f.22 ]
gap> A.size;
12
gap> A.crQClass;
[ 4, 20, 3 ] ```

`CharTableQClass( dim, system, q-class )` `CharTableQClass( dim, IT-number )`
`CharTableQClass( Hermann-Mauguin-symbol )`

`CharTableQClass` returns the character table T, say, of a representative group of (a Z-class of) the specified Q-class.

Although the set of characters can be considered as an invariant of the specified Q-class, the resulting table will depend on the order in which GAP sorts the conjugacy classes of elements and the irreducible characters and hence, in general, will not coincide with the corresponding table presented in BBNWZ78.

`CharTableQClass` proceeds as follows. If the groups in the given Q-class are solvable, then it first calls the `AgGroupQClass` and `RefinedAgSeries` functions to get an isomorphic ag group with a PAG system, and then it calls the `CharTable` function to compute the character table of that ag group. In the case of one of the five Q-classes of dimension 4 whose groups are not solvable, it first calls the `FpGroupQClass` function to get an isomorphic finitely presented group, then it constructs a specially chosen faithful permutation representation of low degree for that group, and finally it determines the character table of the resulting permutation group again by calling the `CharTable` function.

In general, the above strategy will be much more efficient than the alternative possibilities of calling the `CharTable` function for a finitely presented group provided by the `FpGroupQClass` function or for a matrix group provided by the `MatGroupZClass` function.

```    gap> T := CharTableQClass( 4, 20, 3 );;
gap> DisplayCharTable( T );
CharTableQClass( 4, 20, 3 )

2  2  1  1  2  2  2
3  1  1  1  1  .  .

1a 3a 6a 2a 4a 4b
2P 1a 3a 3a 1a 2a 2a
3P 1a 1a 2a 2a 4b 4a
5P 1a 3a 6a 2a 4a 4b

X.1     1  1  1  1  1  1
X.2     1  1  1  1 -1 -1
X.3     1  1 -1 -1  A -A
X.4     1  1 -1 -1 -A  A
X.5     2 -1  1 -2  .  .
X.6     2 -1 -1  2  .  .

A = E(4)
= ER(-1) = i ```

`NrZClassesQClass( dim, system, q-class )` `NrZClassesQClass( dim, IT-number )`
`NrZClassesQClass( Hermann-Mauguin-symbol )`

`NrZClassesQClass` returns the number of Z-classes within the given Q-class. It can be used to formulate loops over the Z-classes.

The following functions are functions of Z-classes.

In general, the parameters characterizing a Z-class will form a quadruple `(dim,,system, mboxq-class,,z-class)` where dim is the associated dimension, system is the number of the associated crystal system, q-class is the number of the associated Q-class within the crystal system, and z-class is the number of the Z-class within the Q-class. However, in case of dimensions 2 or 3, a Z-class may also be characterized by a pair `(dim, IT-number)` where IT-number is the number in the International Tables Hah83 of any space-group type lying in that Z-class, or just by the Hermann-Mauguin symbol of any space-group type lying in that Z-class.

`DisplayZClass( dim, system, q-class, z-class )` `DisplayZClass( dim, IT-number )`
`DisplayZClass( Hermann-Mauguin-symbol )`

`DisplayZClass` displays for the specified Z-class essentially the same information as is provided for that Z-class in Table 1 of BBNWZ78 (except for the generating matrices of a class representative group given there), namely item for dimensions 2 and 3, the Hermann-Mauguin symbol of a representative space-group type which belongs to that Z-class, vspace-2mm item the Bravais type, vspace-2mm item some decomposability information, vspace-2mm item the number of space-group types belonging to the Z-class, vspace-2mm item the size of the associated cohomology group. vspace-2mm For details see BBNWZ78.

```    gap> DisplayZClass( 2, 3 );
#I    Z-class (2,2,1,1) = Z(pm): Bravais type II/I; fully Z-reducible;
#I     2 space groups; cohomology group size 2
gap> DisplayZClass( "F-43m" );
#I    Z-class (3,7,4,2) = Z(F-43m): Bravais type VI/II; Z-irreducible;
#I     2 space groups; cohomology group size 2
gap> DisplayZClass( 4, 2, 3, 2 );
#I    Z-class B (4,2,3,2): Bravais type II/II; Z-decomposable;
#I     2 space groups; cohomology group size 4
gap> DisplayZClass( 4, 21, 3, 1 );
#I   *Z-class (4,21,3,1): Bravais type XVI/I; Z-reducible;
#I     1 space group; cohomology group size 1 ```

`MatGroupZClass( dim, system, q-class, z-class )` `MatGroupZClass( dim, IT-number )`
`MatGroupZClass( Hermann-Mauguin-symbol )`

`MatGroupZClass` returns a dim times dim matrix group M, say, which is a representative of the specified Z-class. Its generators satisfy the defining relators of the finitely presented group which may be computed by calling the `FpGroupQClass` function (see above) for the Q-class which contains the given Z-class.

The generators of M are the same matrices as those given in Table 1 of BBNWZ78. Note, however, that they will be listed in reverse order to keep them in parallel to the abstract generators provided by the `FpGroupQClass` function (see above).

Besides of the usual components, the group record of M will have an additional component M`.crZClass` which saves a list of the parameters that specify the given Z-class. (In fact, in order to make the resulting group record consistent with those returned by the `NormalizerZClass` or `ZClassRepsDadeGroup` functions described below, it also will have an additional component M.`crConjugator` containing just the identity element of M.)

```    gap> M := MatGroupZClass( 4, 20, 3, 1 );
MatGroupZClass( 4, 20, 3, 1 )
gap> for g in M.generators do
>  Print( "\n" ); PrintArray( g ); od; Print( "\n" );

[ [   0,   1,   0,   0 ],
[  -1,   0,   0,   0 ],
[   0,   0,  -1,  -1 ],
[   0,   0,   0,   1 ] ]

[ [  -1,   0,   0,   0 ],
[   0,  -1,   0,   0 ],
[   0,   0,  -1,  -1 ],
[   0,   0,   1,   0 ] ]

gap> M.size;
12
gap> M.crZClass;
[ 4, 20, 3, 1 ] ```

`NormalizerZClass( dim, system, q-class, z-class )` `NormalizerZClass( dim, IT-number )`
`NormalizerZClass( Hermann-Mauguin-symbol )`

`NormalizerZClass` returns the normalizer N, say, in GL(dim,Z) of the representative dim times dim matrix group which is constructed by the `MatGroupZClass` function (see above).

If the size of N is finite, then N again lies in some Z-class. In this case, the group record of N will contain two additional components N`.crZClass` and N.`crConjugator` which provide the parameters of that Z-class and a matrix g in GL(dim,Z), respectively, such that N = g^{-1} R g, where R is the representative group of that Z-class.

```    gap> N := NormalizerZClass( 4, 20, 3, 1 );
NormalizerZClass( 4, 20, 3, 1 )
gap> for g in N.generators do
>  Print( "\n" ); PrintArray( g ); od; Print( "\n" );

[ [   1,   0,   0,   0 ],
[   0,   1,   0,   0 ],
[   0,   0,   1,   0 ],
[   0,   0,  -1,  -1 ] ]

[ [   1,   0,   0,   0 ],
[   0,  -1,   0,   0 ],
[   0,   0,  -1,  -1 ],
[   0,   0,   1,   0 ] ]

[ [   0,   1,   0,   0 ],
[  -1,   0,   0,   0 ],
[   0,   0,   1,   0 ],
[   0,   0,   0,   1 ] ]

[ [  -1,   0,   0,   0 ],
[   0,  -1,   0,   0 ],
[   0,   0,  -1,   0 ],
[   0,   0,   0,  -1 ] ]

gap> N.size;
96
gap> N.crZClass;
[ 4, 20, 22, 1 ]
gap> N.crConjugator = N.identity;
true ```

```    gap> L := NormalizerZClass( 3, 42 );
NormalizerZClass( 3, 3, 2, 4 )
gap> L.size;
16
gap> L.crZClass;
[ 3, 4, 7, 2 ]
gap> L.crConjugator;
[ [ 0, 0, -1 ], [ 1, 0, 0 ], [ 0, -1, -1 ] ]
gap> M := NormalizerZClass( "C2/m" );
Group( [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ],
[ [ 0, -1, 0 ], [ -1, 0, 0 ], [ 0, 0, -1 ] ],
[ [ 1, 0, 1 ], [ 0, 1, 1 ], [ 0, 0, 1 ] ],
[ [ -1, 0, 0 ], [ 0, -1, 0 ], [ -1, -1, 1 ] ],
[ [ 0, 1, -1 ], [ 1, 0, -1 ], [ 0, 0, -1 ] ] )
gap> M.size;
"infinity"
gap> IsBound( M.crZClass );
false ```

`NrSpaceGroupTypesZClass( dim, system, q-class, z-class )` `NrSpaceGroupTypesZClass( dim, IT-number )`
`NrSpaceGroupTypesZClass( Hermann-Mauguin-symbol )`

`NrSpaceGroupTypes` returns the number of space-group types within the given Z-class. It can be used to formulate loops over the space-group types.

```    gap> N := NrSpaceGroupTypesZClass( 4, 4, 1, 1 );
13 ```

Some of the Z-classes of dimension d, say, are ``maximal'' in the sense that the groups in these classes are maximal finite subgroups of GL(d,Z). Generalizing a term which is being used for dimension 4, we call the representatives of these maximal Z-classes the ``Dade groups'' of dimension d.

`NrDadeGroups( dim )`

`NrDadeGroups` returns the number of Dade groups of dimension dim. It can be used to formulate loops over the Dade groups.

There are 2, 4, and 9 Dade groups of dimension 2, 3, and 4, respectively.

```    gap> NrDadeGroups( 4 );
9 ```

`DadeGroup( dim, n )`

`DadeGroup` returns the nth Dade group of dimension dim.

```    gap> D := DadeGroup( 4, 7 );
MatGroupZClass( 4, 31, 7, 2 ) ```

`DadeGroupNumbersZClass( dim, system, q-class, z-class )` `DadeGroupNumbersZClass( dim, IT-number )`
`DadeGroupNumbersZClass( Hermann-Mauguin-symbol )`

`DadeGroupNumbersZClass` returns the set of all those integers n_i for which the n_ith Dade group of dimension dim contains a subgroup which, in GL(dim,Z), is conjugate to the representative group of the given Z-class.

```    gap> dadeNums := DadeGroupNumbersZClass( 4, 4, 1, 2 );
[ 1, 5, 8 ]
gap> for d in dadeNums do
>     D := DadeGroup( 4, d );
>     Print( D, " of size ", Size( D ), "\n" );
> od;
MatGroupZClass( 4, 20, 22, 1 ) of size 96
MatGroupZClass( 4, 30, 13, 1 ) of size 288
MatGroupZClass( 4, 32, 21, 1 ) of size 384 ```

`ZClassRepsDadeGroup( dim, system, q-class, z-class, n )` `ZClassRepsDadeGroup( dim, IT-number, n )`
`ZClassRepsDadeGroup( Hermann-Mauguin-symbol, n )`

`ZClassRepsDadeGroup` determines in the nth Dade group of dimension dim all those conjugacy classes whose groups are, in GL(dim,Z), conjugate to the Z-class representative group R, say, of the given Z-class. It returns a list of representative groups of these conjugacy classes.

Let M be any group in the resulting list. Then the group record of M provides two components M`.crZClass` and M`.crConjugator` which contain the list of Z-class parameters of R and a suitable matrix g from GL(dim,Z), respectively, such that M equals g^{-1} R g.

```    gap> DadeGroupNumbersZClass( 2, 2, 1, 2 );
[ 1, 2 ]
gap> ZClassRepsDadeGroup( 2, 2, 1, 2, 1 );
[ MatGroupZClass( 2, 2, 1, 2 )^[ [ 0, 1 ], [ -1, 0 ] ] ]
gap> ZClassRepsDadeGroup( 2, 2, 1, 2, 2 );
[ MatGroupZClass( 2, 2, 1, 2 )^[ [ 1, -1 ], [ 0, -1 ] ],
MatGroupZClass( 2, 2, 1, 2 )^[ [ 1, 0 ], [ -1, 1 ] ] ]
gap> R := last[2];;
gap> R.crZClass;
[ 2, 2, 1, 2 ]
gap> R.crConjugator;
[ [ 1, 0 ], [ -1, 1 ] ] ```

The following functions are functions of space-group types.

In general, the parameters characterizing a space-group type will form a quintuple `(dim, system,,q-class,,z-class,,sg-type)` where dim is the associated dimension, system is the number of the associated crystal system, q-class is the number of the associated Q-class within the crystal system, z-class is the number of the Z-class within the Q-class, and sg-type is the space-group type within the Z-class. However, in case of dimensions 2 or 3, you may instead specify a Z-class by a pair `(dim, IT-number)` or by its Hermann-Mauguin symbol (as described above). Then the function will handle the first space-group type within that Z-class, i.,e., sg-type = 1, that is, the corresponding symmorphic space group (split extension).

```DisplaySpaceGroupType( dim, system, q-class, z-class, sg-type )``` `DisplaySpaceGroupType( dim, IT-number )`
`DisplaySpaceGroupType( Hermann-Mauguin-symbol )`

`DisplaySpaceGroupType` displays for the specified space-group type some of the information which is provided for that space-group type in Table 1 of BBNWZ78, namely item the orbit size associated with that space-group type and, vspace-2mm item for dimensions 2 and 3, the IT-number and the Hermann-Mauguin symbol. vspace-2mm For details see BBNWZ78.

```    gap> DisplaySpaceGroupType( 2, 17 );
#I     Space-group type (2,4,4,1,1); IT(17) = p6mm; orbit size 1
gap> DisplaySpaceGroupType( "Pm-3" );
#I     Space-group type (3,7,2,1,1); IT(200) = Pm-3; orbit size 1
gap> DisplaySpaceGroupType( 4, 32, 10, 2, 4 );
#I    *Space-group type (4,32,10,2,4); orbit size 18
gap> DisplaySpaceGroupType( 3, 6, 1, 1, 4 );
#I    *Space-group type (3,6,1,1,4); IT(169) = P61, IT(170) = P65;
#I      orbit size 2; fp-free ```

```DisplaySpaceGroupGenerators( dim, system, q-class, z-class, sg-type )``` `DisplaySpaceGroupGenerators( dim, IT-number )`
`DisplaySpaceGroupGenerators( Hermann-Mauguin-symbol )`

`DisplaySpaceGroupGenerators` displays the non-translation generators of a representative space group of the specified space-group type without actually constructing that matrix group.

In more details: Let n = dim be the given dimension, and let M_1, ldots, M_r be the generators of the representative n times n matrix group of the given Z-class (this is the group which you will get if you call the `MatGroupZClass` function (see above) for that Z-class). Then, for the given space-group type, the `SpaceGroup` function described below will construct as representative of that space-group type an (n+1) times (n+1) matrix group which is generated by the n translations which are induced by the (standard) basis vectors of the n-dimensional Euclidian space, and r additional matrices S_1, ldots, S_r of the form S_i = left[ catcode`|=12 begin{tabular}{c|c} catcode`|=13 M_i & t_i \ hline 0 & 1 end{tabular} right], where the n times n submatrices M_i are as defined above, and the t_i are n-columns with rational entries. The `DisplaySpaceGroupGenerators` function saves time by not constructing the group, but just displaying the r matrices S_1, ldots, S_r.

```    gap> DisplaySpaceGroupGenerators( "P61" );
#I  The non-translation generators of SpaceGroup( 3, 6, 1, 1, 4 ) are

[ [   -1,    0,    0,    0 ],
[    0,   -1,    0,    0 ],
[    0,    0,    1,  1/2 ],
[    0,    0,    0,    1 ] ]

[ [    0,   -1,    0,    0 ],
[    1,   -1,    0,    0 ],
[    0,    0,    1,  1/3 ],
[    0,    0,    0,    1 ] ] ```

`SpaceGroup( dim, system, q-class, z-class, sg-type )` `SpaceGroup( dim, IT-number )`
`SpaceGroup( Hermann-Mauguin-symbol )`

`SpaceGroup` returns a (dim+1) times (dim+1 ) matrix group S, say, which is a representative of the given space-group type (see also the description of the `DisplaySpaceGroupGenerators` function above).

```    gap> S := SpaceGroup( "P61" );
SpaceGroup( 3, 6, 1, 1, 4 )
gap> for s in S.generators do
>  Print( "\n" ); PrintArray( s ); od; Print( "\n" );

[ [   -1,    0,    0,    0 ],
[    0,   -1,    0,    0 ],
[    0,    0,    1,  1/2 ],
[    0,    0,    0,    1 ] ]

[ [    0,   -1,    0,    0 ],
[    1,   -1,    0,    0 ],
[    0,    0,    1,  1/3 ],
[    0,    0,    0,    1 ] ]

[ [  1,  0,  0,  1 ],
[  0,  1,  0,  0 ],
[  0,  0,  1,  0 ],
[  0,  0,  0,  1 ] ]

[ [  1,  0,  0,  0 ],
[  0,  1,  0,  1 ],
[  0,  0,  1,  0 ],
[  0,  0,  0,  1 ] ]

[ [  1,  0,  0,  0 ],
[  0,  1,  0,  0 ],
[  0,  0,  1,  1 ],
[  0,  0,  0,  1 ] ]

gap> S.crSpaceGroupType;
[ 3, 6, 1, 1, 4 ] ```

Besides of the usual components, the resulting group record of S contains an additional component S`.crSpaceGroupType` which saves a list of the parameters that specify the given space-group type.

Moreover, it contains, in form of a finitely presented group, a presentation of S which is satisfied by the matrix generators. If the factor group of S by its translation normal subgroup is solvable then this presentation is chosen such that it is a polycyclic power commutator presentation. The proper way to access this presentation is to call the following function.

`FpGroup( S )`

`FpGroup` returns a finitely presented group G, say, which is isomorphic to S, where S is expected to be a space group. It is chosen such that there is an isomrphism from G to S which maps each generator of G onto the corresponding generator of S. This means, in particular, that the matrix generators of S satisfy the relators of G.

```    gap> G := FpGroup( S );
Group( g1, g2, g3, g4, g5 )
gap> for rel in G.relators do Print( rel, "\n" ); od;
g1^2*g5^-1
g2^3*g5^-1
g2^-1*g1^-1*g2*g1
g3^-1*g1^-1*g3*g1*g3^2
g3^-1*g2^-1*g3*g2*g4*g3^2
g4^-1*g1^-1*g4*g1*g4^2
g4^-1*g2^-1*g4*g2*g4*g3^-1
g4^-1*g3^-1*g4*g3
g5^-1*g1^-1*g5*g1
g5^-1*g2^-1*g5*g2
g5^-1*g3^-1*g5*g3
g5^-1*g4^-1*g5*g4
gap> # Verify that the matrix generators of S satisfy the relators of G.
gap> ForAll( G.relators,
>  rel -> MappedWord( rel, G.generators, S.generators ) = S.identity );
true ```

`TransposedSpaceGroup( dim, system, q-class, z-class, sg-type )` `TransposedSpaceGroup( dim, IT-number )`
`TransposedSpaceGroup( Hermann-Mauguin-symbol )`
`TransposedSpaceGroup( S )`

`TransposedSpaceGroup` returns a matrix group T, say, whose generators are just the transposed generators (in the same order) of the corresponding space group S specified by the arguments. As for S, you may get a finite presentation for T via the `FpGroup` function.

The purpose of this function is explicitly discussed in the introduction to this section.

```    gap> T := TransposedSpaceGroup( S );
TransposedSpaceGroup( 3, 6, 1, 1, 4 )
gap> for m in T.generators do
>  Print( "\n" ); PrintArray( m ); od; Print( "\n" );

[ [   -1,    0,    0,    0 ],
[    0,   -1,    0,    0 ],
[    0,    0,    1,    0 ],
[    0,    0,  1/2,    1 ] ]

[ [    0,    1,    0,    0 ],
[   -1,   -1,    0,    0 ],
[    0,    0,    1,    0 ],
[    0,    0,  1/3,    1 ] ]

[ [  1,  0,  0,  0 ],
[  0,  1,  0,  0 ],
[  0,  0,  1,  0 ],
[  1,  0,  0,  1 ] ]

[ [  1,  0,  0,  0 ],
[  0,  1,  0,  0 ],
[  0,  0,  1,  0 ],
[  0,  1,  0,  1 ] ]

[ [  1,  0,  0,  0 ],
[  0,  1,  0,  0 ],
[  0,  0,  1,  0 ],
[  0,  0,  1,  1 ] ] ```

## 37.14 The Small Groups Library

This library contains all groups of order at most 1000 except for 512 and 768 up to isomorphism. There are a total of 174366 such groups.

`SmallGroup( size, i )`

The function `SmallGroup( size, i )` returns the ith group of order size in the catalogue. It will return an AgGroup, if the group is soluble and a PermGroup otherwise.

`NumberSmallGroups( size )`

The function `NumberSmallGroups( size )` returns the number of groups of the order size.

`AllSmallGroups( size )`

The function `AllSmallGroups( size )` returns the list of all groups of the order size.

`UnloadSmallGroups( list of sizes )`

It is possible to work with the catalogue of groups of small order just using the functions described above. However, the catalogue is rather large even though the groups are stored in a very compact description. Thus it might be helpful for a space efficient usage of the catalogue, to know a little bit about unloading parts of the catalogue by hand.

At the first call of one of the functions described above, the groups of order size are loaded and stored in a compact description. GAP will not unload them itsself again. Thus if one calls one of the above functions for a lot of different orders, then all the groups of these orders are stored. Even though the description of the groups is space efficient, this might use a lot of space. For example, if one uses the above functions to load the complete catalogue, then GAP will grow to about 12 MB of workspace.

Thus it might be interesting to unload the groups of some orders again, if they are not used anymore. This can be done by calling the function `UnloadSmallGroups( list of sizes )`

If the groups of order size are unloaded by hand, then GAP will of course load them again at the next call of `SmallGroup( size, i )` or one of the other functions described at the beginning of this section.

`IdGroup( G )`

Let G be a PermGroup or AgGroup of order at most 1000, but not of order 256, 512 or 768. Then the function call `IdGroup( G )` returns a tuple [size, i] meaning that G is isomorphic to the i-th group in the catalogue of groups of order size.

Note that this package calls and uses the ANUPQ share library of GAP in a few cases.

GAP 3.4.4
April 1997