60 CrystGap--The Crystallographic Groups Package

The CrystGap package provides functions for the computation with affine crystallographic groups, in particular space groups. Also provided are some functions dealing with related linear matrix groups, such as point groups. For the definition of the standard crystallographic notions we refer to the International Tables Hah95, in particular the chapter by Wondratschek Won95, and to the introductory chapter in Irreducible Maximal Finite Integral Matrix Groups. The principal algorithms used in this package are described in EGN97, a preprint of which in included in the doc directory of this package.

CrystGap is implemented in the GAP language, and runs on any system supporting GAP~3.4.4. The function WyckoffLattice, however, requires the share package sf XGap, which in turn runs only under Unix. The functions described in this chapter can be used only after loading CrystGap with the command

 
     gap> RequirePackage( "cryst" ); 

CrystGap has been developed by

Bettina Eick

:
Lehrstuhl D faccent127 ur Mathematik, RWTH Aachen, D-52056 Aachen, Germany
:e-mail: Bettina.Eick@math.RWTH-Aachen.de

Franz Gaccent127 ahler

:
Centre de Physique Thaccent19 eorique, Ecole Polytechnique, F-91128 Palaiseau, France
:e-mail: gaehler@pth.polytechnique.fr

Werner Nickel

:
School of Mathematical and Computational Sciences, University of St~Andrews,
:St~Andrews, Fife KY16 9SS, Scotland
:e-mail: werner@dcs.st-and.ac.uk

Please send bug reports, suggestions and other comments to any of these e-mail addresses.

The first and third authors acknowledge financial support from the Graduiertenkolleg it Ana-lyse und Konstruktion in der Mathematik. The second author was supported by the Swiss Bundesamt faccent127 ur Bildung und Wissenschaft in the framework of the HCM programme of the European Community. This collaboration was in part made possible by financial support from the HCM project it Computational Group Theory.

Subsections

  1. Crystallographic Groups
  2. Space Groups
  3. More about Crystallographic Groups
  4. CrystGroup
  5. IsCrystGroup
  6. PointGroup
  7. TranslationsCrystGroup
  8. AddTranslationsCrystGroup
  9. CheckTranslations
  10. ConjugatedCrystGroup
  11. FpGroup for point groups
  12. FpGroup for CrystGroups
  13. MaximalSubgroupsRepresentatives
  14. IsSpaceGroup
  15. IsSymmorphicSpaceGroup
  16. SpaceGroupsPointGroup
  17. Wyckoff Positions
  18. WyckoffPositions
  19. WyckoffPositionsByStabilizer
  20. WyckoffPositionsQClass
  21. WyckoffOrbit
  22. WyckoffLattice
  23. NormalizerGL
  24. CentralizerGL
  25. PointGroupsBravaisClass
  26. TranslationNormalizer
  27. AffineNormalizer
  28. AffineInequivalentSubgroups
  29. Other functions for CrystGroups
  30. Color Groups
  31. ColorGroup
  32. IsColorGroup
  33. ColorSubgroup
  34. ColorCosets
  35. ColorOfElement
  36. ColorPermGroup
  37. ColorHomomorphism
  38. Subgroup for color groups
  39. PointGroup for color CrystGroups
  40. Inequivalent colorings of space groups

60.1 Crystallographic Groups

An affine crystallographic group G is a subgroup of the group of all Euclidean motions of d-dimensional space, with the property that its subgroup T of all pure translations is a freely abelian, normal subgroup of G, which has rank at most equal to d, and which has finite index in G. In this package, the term CrystGroup always refers to such an affine crystallographic group. Linear matrix groups, whether crystallographic or not, will carry different designations (see below). CrystGroups are represented as special matrix groups, whose elements are affine matrices of the form

                     [ M 0 ]
                     [ t 1 ]
acting on row vectors (x,1) from the right. Note that this is different from the crystallographic convention, where matrices usually act from the left on column vectors (see also The Crystallographic Groups Library). We have adopted this convention to maintain compatibility with the rest of GAP.

The ``linear" parts M of the elements of a CrystGroup G generate the point group P of G, which is isomorphic to the quotient G/T. There is a natural homomorphism from G to P, whose kernel is T. The translation vectors of the elements of T generate a free Z-module L, called the translation lattice of G. CrystGroups can be defined with respect to any basis of Euclidean space, but internally most computations will be done in a basis which contains a basis of L (see More about Crystallographic Groups).

CrystGroups carry a special operations record CrystGroupOps, and are identified with a tag isCrystGroup. CrystGroups must be constructed with a call to CrystGroup (see CrystGroup) which sets the tag isCrystGroup to true, and sets the operations record to CrystGroupOps.

Warning: The groups in GAP' s crystallographic groups library (see The Crystallographic Groups Library), whether they are extracted with SpaceGroup or TransposedSpaceGroup, are not CrystGroups in the sense of this package, because CrystGroups have different record entries and a different operations record. However, a group extracted with TransposedSpaceGroup from that library can be converted to a CrystGroup by a call to CrystGroup (see CrystGroup).

60.2 Space Groups

A CrystGroup which has a translation subgroup of full rank is called a space group. Certain functions are available only for space groups, and not for general CrystGroups, notably all functions dealing with Wyckoff positions (see Wyckoff Positions).

Space groups which are equivalent under conjugation in the affine group (shortly: affine equivalent space groups) are said to belong to the same space group type. As is well known, in three dimensions there are 219 such space group types (if only conjugation by transformations with positive determinant is allowed, there are 230).

Representatives of all space group types in dimensions 2, 3 and 4 can be obtained from the crystallographic groups library contained in GAP (see The Crystallographic Groups Library). They must be extracted with the function CrystGroup, and not with the usual extraction functions SpaceGroup and TransposedSpaceGroup of that library, as these latter functions return groups which do not have an operations record that would allow to compute with them. CrystGroup accepts exactly the same arguments as SpaceGroup and TransposedSpaceGroup. It returns the same group as TransposedSpaceGroup, but equipped with a working operations record.

Space group types (and thus space groups) are classified into Z-classes and Q-classes. Two space groups belong to the same Z-class if their point groups, expressed in a basis of their respective translation lattices, are conjugate as subgroups of GL(d,Z). If the point groups are conjugate as subgoups of GL(d,Q), the two space groups are said to be in the same Q-class. This provides also a classification of point groups (expressed in a lattice basis, i.e., integral point groups) into Z-classes and Q-classes.

For a given finite integral matrix group P, representing a point group expressed in a lattice basis, a set of representative space groups for each space group type in the Z-class of P can be obtained with SpaceGroupsPointGroup (see SpaceGroupsPointGroup). If, moreover, the normalizer of P in GL(d,Z) is known (see NormalizerGL), exactly one representative is obtained for each space group type. Representatives of all Z-classes of maximal irreducible finite point Irreducible Maximal Finite Integral Matrix Groups) in all dimensions up to 11, and in prime dimensions up to 23. For some other dimensions, at least Q-class representatives are available. Important information about a space group is contained in its affine normalizer (see AffineNormalizer), which is the normlizer of the space group in the affine group. In a way, the affine normalizer can be regarded as the symmetry of the space group.

Warning: Groups which are called space groups in this manual should not be confused with groups extracted with SpaceGroup from the The Crystallographic Groups Library). The latter are not CrystGroups in the sense of this package.

60.3 More about Crystallographic Groups

In this section we describe how a CrystGroup G is represented internally. The casual user can skip this section in a first reading. Although the generators of a CrystGroup can be specified with respect to any basis, most computations are done internally in a special, standard basis, which is stored in G.internalBasis. The results are translated into the user-specified basis only afterwards. G.internalBasis consists of a (standard) basis of the translation lattice of G, complemented, if necessary, with suitable standard basis vectors. The standard basis of the translation lattice is stored in G.translations.

As soon as G.internalBasis has been determined, both the CrystGroup G and its point group P obtain a component internalGenerators. For the point group P, the component P.internalGenerators contains a set of generators of P, expressed with respect to the internalBasis of G, whereas for the CrystGroup G the component G.internalGenerators contains a set of homomorphic preimages of P.internalGenerators in G, also expressed in the internalBasis of G. Thus G.internalGenerators does not contain any translation generators. These are easy to add, however: With respect to the internal basis, the translations are generated by the first k standard basis vectors, where k is the rank of the translation lattice.

Note that the internalGenerators of both a point group P and a CrystGroup G may be changed by some functions, notably by FpGroup. Thus they need not have any obvious connection to P.generators and G.generators, respectively. Internal record entries of a CrystGroup should never be changed by the user.

60.4 CrystGroup

CrystGroup( matgroup )

CrystGroup( generating matrices )

CrystGroup( list of generators, identity )

CrystGroup( integers )

CrystGroup( string )

CrystGroup accepts as arguments either a group of affine matrices, or a list of generating affine matrices, or an argument identifying a space group from the crystallographic groups library, i.e., a list of two or five integers, or a string containing a Hermann-Mauguin symbol, and converts it into a CrystGroup in the sense of this package. CrystGroup tests whether the generators are indeed affine matrices.

60.5 IsCrystGroup

IsCrystGroup( G )

tests whether G.isCrystGroup is present and true. G.isCrystGroup is set by CrystGroup.

60.6 PointGroup

PointGroup( G )

extracts the point group P of a space group G, binds it to G.pointGroup, and returns it. It also determines the homomorphism from G to P, and binds it to G.pointHomom. A point group P has always a component P.isPointGroup set to true, and a component P.crystGroup containing the CrystGroup from which it was constructed.

60.7 TranslationsCrystGroup

TranslationsCrystGroup( G )

determines a basis of the translation lattice of G, binds it to G.translations, and returns it. Note that this translation lattice is always invariant under the point group P of G. If G.translations is not yet present, a finite presentation of P needs to be determined. A basis of the translation lattice can also be added by the user, with AddTranslationsCrystGroup (see AddTranslationsCrystGroup).

Warning: The component G.translations must never be set by hand. The functions TranslationsCrystGroup and AddTranslationsCrystGroups have important (and wanted) side effects.

60.8 AddTranslationsCrystGroup

AddTranslationsCrystGroup( G, basis )

Since TranslationsCrystGroup (see TranslationsCrystGroup) needs a presentation of the point group, the computation of G.translations can be rather time consuming. If a basis of the translation lattice is known, AddTranslationsCrystGroup can be used to add this knowledge to a CrystGroup. If G.translations is already known, its value is kept without further notice. It is the responsibility of the user that the basis handed over to AddTranslationsCrystGroup is a correct basis of the translation lattice. In case of doubt, the function CheckTranslations (see CheckTranslations) can be used to check whether the basis added was indeed correct.

Warning: The component G.translations must never be set by hand. The functions TranslationsCrystGroup and AddTranslationsCrystGroups have important (and wanted) side effects.

60.9 CheckTranslations

CheckTranslations( G )

checks whether G.translations is indeed correct. If G.translations is incorrect, a warning message is printed, otherwise GAP remains silent. In the case of an incorrect translation basis a new CrystGroup must be created, and the computations must be started afresh, because the wrong translation basis may have produced wrong information components. CheckTranslations is useful if a basis has been added with AddTranslationsCrystGroup, and doubts arise later whether the basis added was correct.

60.10 ConjugatedCrystGroup

ConjugatedCrystGroup( G, c )

returns a new CrystGroup which is a conjugate of G. The conjugator c can either be a d-dimensional linear matrix (which then is complemented with the zero translation), or a (d+1)-dimensional affine matrix. The generators are conjugated as g^c = c*g*c^{-1}. Some components which are bound in G are copied and translated to the new basis, in particular G.generators, G.translations, G.internalBasis, and G.wyckoffPositons. If G.internalBasis is bound,

ConjugatedCrystGroup( G, G.internalBasis )

returns a CrystGroup whose translation lattice (of rank k) is generated by the first k rows of the identity matrix. ConjugatedCrystGroup allows as input only a parent CrystGroup.

60.11 FpGroup for point groups

FpGroup( P )

computes a finite presentation of the point group P, and binds it to P.fpGroup. If P (and thus its CrystGroup G := P.crystGroup) is solvable, a power-commutator presentation is returned.

Warning: If P is solvable, the abstract generators are not More about Crystallographic Groups).

60.12 FpGroup for CrystGroups

FpGroup( G )

computes a finite presentation of the CrystGroup G, and binds it to G.fpGroup. If the point group (and thus G) is solvable, a power-commutator presentation is returned. The presentation is always an extension of the presentation of the point group (which is computed if necessary).

Warning: The abstract generators of the presentation are not More about Crystallographic Groups).

60.13 MaximalSubgroupsRepresentatives

MaximalSubgroupsRepresentatives( S, "translationEqual", [, ind ] )

MaximalSubgroupsRepresentatives( S, "classEqual", ind )

MaximalSubgroupsRepresentatives( S, ind )

returns a list of conjugacy class representatives of maximal subgroups of the CrystGroup S. If ind is present, which must be a prime or a list of primes, only those subgroups are returned whose index is a power of a prime contained in or equal to ind. If the flag ``translationEqual" is present, only those subgroups are returned which are translation-equal (translationengleich) with S. If the flag ``classEqual" is present, only those subgroups are return which are class-equal (klassengleich) with S. ind is optional only if the flag ``latticeEqual" is present. In all other cases, ind is required.

60.14 IsSpaceGroup

IsSpaceGroup( S )

determines whether the CrystGroup S is a space group (see Crystallographic Groups).

60.15 IsSymmorphicSpaceGroup

IsSymmorphicSpaceGroup( S )

determines whether the space group S is symmorphic. A space group is called symmorphic if it is equivalent to a semidirect product of its point group with its translation subgroup.

60.16 SpaceGroupsPointGroup

SpaceGroupsPointGroup( P )

SpaceGroupsPointGroup( P, normalizer elements )

where P is any finite subgroup of GL(d,Z), returns a list of all space groups with point group P, up to conjugacy in the full translation group of Euclidean space. All these space groups are returned as CrystGroups in standard representation. If a second argument is present, which must be a list of elements of the normalizer of P in GL(d,Z), only space groups inequivalent under conjugation with these elements are returned. If these normalizer elements, together with P, generate the full normalizer of P in GL(d,Z), then exactly one representative of each space group type is obtained.

60.17 Wyckoff Positions

A Wyckoff position of a space group G is an equivalence class of points in Euclidean space, having stabilizers which are conjugate subgroups of G. Apart from a subset of lower dimension, which contains points with even bigger stabilizers, a Wyckoff position consists of a G-orbit of some affine subspace A. A Wyckoff position W therefore can be specified by a representative affine subspace A and its stabilizer subgroup. In CrystGap, a Wyckoff position W is represented as a record with the following components:

W.basis:

Basis of the linear space L parallel to A. This basis is also a basis of the intersection of L with the translation lattice of S.
Can be extracted with WyckoffBasis( W ).

W.translation:

W.translation is such that A = L + W.translation.
Can be extracted with WyckoffTranslation( W ).

W.stabilizer:

The stabilizer subgroup of any generic point in A.
Can be extracted with WyckoffStabilizer( W ).

W.class:

Wyckoff positions carry the same class label if and only if their stabilizers have point groups which are conjugate subgroups of the point group of S.
Can be extracted with WyckoffPosClass( W ).

W.spaceGroup:

The space group of which it is a Wyckoff position.
Can be extracted with WyckoffSpaceGroup( W ).

W.isWyckoffPosition:

A flag identifying the record as a Wyckoff position. It is set to true.
Can be tested with IsWyckoffPosition( W ).

W.operations:

The operations record of a Wyckoff position. It currently contains only a Print function.

60.18 WyckoffPositions

WyckoffPositions( G )

returns the list of all Wyckoff positions of the space group G.

60.19 WyckoffPositionsByStabilizer

WyckoffPositionsByStabilizer( G, U ),

where G is a space group and U a subgroup of the point group or a list of such subgroups, determines only the Wyckoff positions (see WyckoffPositions) having a representative affine subspace whose stabilizer has a point group equal to the subgroup U or contained in the list U, respectively.

60.20 WyckoffPositionsQClass

WyckoffPositionsQClass( G, S )

For space groups with larger point groups, most of the time in the computation of Wyckoff positions (see WyckoffPositions) is spent computing the subgroup lattice of the point group. If Wyckoff positions are needed for several space groups which are in the same Q class, and therefore have isomorphic point groups, one can avoid recomputing the same subgroup lattice for each of them as follows. For the computation of the Wyckoff positions of the first space group S one uses a call to WyckoffPositions. For the remaining space groups, S is then passed as a second argument to WyckoffPositionsQClass( G, S ), which uses some of the results already obtained for S.

60.21 WyckoffOrbit

WyckoffOrbit( W )

takes a Wyckoff position W (see Wyckoff Positions) and returns a list of Wyckoff positions which are different representations of W, such that the representative affine subspaces of these representations form an orbit under the space group G of W, modulo lattice translations.

60.22 WyckoffLattice

WyckoffLattice( G )

If a point x in a Wyckoff position W_1 has a stabilizer which is a subgroup of the stabilizer of some point y in a Wyckoff position W_2, then the closure of W_1 will contain W_2. These incidence relations are best represented in a graph. WyckoffLattice( G ) determines and displays this graph using XGAP (note that XGAP runs only under Unix plus the X Window System). Each Wyckoff position is represented by a vertex. If W_1 contains W_2, its vertex is placed below that of W_2 (i.e., Wyckoff positions with bigger stabilizers are placed higher up), and the two are connected, either directly (if there is no other Wyckoff position in between) or indirectly. With the left mouse button and with the XGAP CleanUp menu it is possible to change the layout of the graph (see the XGAP manual). When clicking with the right mouse button on a vertex, a pop up menu appears, which allows to obtain the following information about the representative affine subspace of the Wyckoff position:

StabDim::

Dimension of the affine subspace of stable points.

StabSize::

Size of the stabilizer subgroup.

ClassSize::

Number of Wyckoff positions having a stabilizer whose point group is in the same subgroup conjugacy class.

IsAbelian, IsCyclic, IsNilpotent, IsPerfect, IsSimple,

IsSolvable::

Information about the stabilizer subgroup.

Isomorphism::

Isomorphism type of the stabilizer subgroup. Works only for small sizes.

ConjClassInfo::

Prints (in the GAP window) information about each of the conjugacy classes of the stabilizer, namely the order, the trace and the determinant of its elements, and the size of the conjugacy class. Note that trace refers here only to the trace of the point group part, without the trailing 1 of the affine matrix.

Translation::

The representative point of the affine subspace.

Basis::

The basis of the linear space parallel to the affine subspace.

60.23 NormalizerGL

NormalizerGL( G ),

where G is a finite subgroup of GL(d,Z), returns the normalizer of G in GL(d,Z). At present, this function is available only for groups which are the point group of a CrystGroup extracted from the space group library.

60.24 CentralizerGL

CentralizerGL( G ),

where G is a finite subgroup of GL(d,Z), returns the centralizer of G in GL(d,Z). At present, this function is available only for groups which are the point group of a CrystGroup extracted from the space group library.

60.25 PointGroupsBravaisClass

PointGroupsBravaisClass( B )

PointGroupsBravaisClass( B [, norm ] )

where B is a finite integral matrix group, returns a list of representatives of those conjugacy classes of subgroups of B which are in the same Bravais class as B. These representatives are returned as parent groups, not subgroups. If B is a Bravais group, the list contains a representative of each point group in the Bravais class of B. If a second argument is present, which must be a list of elements of the normalizer of B in GL(d,Z), only subgroups inequivalent under conjugation with these elements are returned.

60.26 TranslationNormalizer

TranslationNormalizer( S )

returns the normalizer of the space group S in the full translation group. At present, this function is implemented only for space groups, not for general CrystGroups. The translation normalizer TN of S may contain a continuous subgroup C. A basis of the space of such continuous translations is bound in TN.continuousTranslations. Since this subgroup is not finitely generated, it is not contained in the group generated by TN.generators. Properly speaking, the translation normalizer is the span of TN and C together.

60.27 AffineNormalizer

AffineNormalizer( S )

returns the affine normalizer of the space group S. The affine normalizer contains the translation normalizer as a subgroup. Similarly as with TranslationNormalizer, the subgroup C of continuous translations, which is not finitely generated, is not part of the group that is returned. However, a basis of the space of continuous translations is bound in the component continuousTranslations.

At present, this function is available only for for space groups, not for general CrystGroups. Moreover, the NormalizerGL (see NormalizerGL) of the point group of S must be know, which currently is the case only for CrystGroups extracted from the space group library.

60.28 AffineInequivalentSubgroups

AffineInequivalentSubgroups( sub )

takes as input a list of subgroups with common parent space group S, and returns a sublist of those which are affine inequivalent. For this, the affine normalizer of S is required, which currently is available only if S is a space group extracted from the space groups library.

60.29 Other functions for CrystGroups

In the operations record of a CrystGroup many of the usual GAP functions are replaced with a CrystGroup specific implementation. For other functions the default implementation can be used. Since CrystGroups are matrix groups, all functions which work for a finite matrix group should work also for a finite CrystGroup (i.e., one which contains no pure translations). Of course, functions which require a finite group as input will work only for finite CrystGroups. Following is a (probably not exhaustive) list of functions that are known to work for also for infinite CrystGroups.

     in
     Parent, IsParent, Group, IsGroup
     Subgroup, IsSubgroup, AsSubgroup, Index 
     Centralizer, Centre, Normalizer 
     Closure, NormalClosure
     Intersection, NormalIntersection
     ConjugacyClassSubgroups, ConjugateSubgroups
     DerivedSubgroup, CommutatorSubgroup, Core 
     DerivedSeries, SubnormalSeries
     FactorGroup, CommutatorFactorGroup
     ConjugateSubgroup, TrivialSubgroup 
     IsAbelian, IsCentral, IsTrivial
     IsNormal, IsSubnormal, IsPerfect, IsSolvable 

The following functions work for CrystGroups provided the subgroup H has finite index in G. The elements of the resulting domain are given in ascending order (with respect to an ad hoc, but fixed ordering).

     Cosets( G, H )
     RightCosets( G, H )
     LeftCosets( G, H )

The following functions dealing with group operations work for CrystGroups provided the orbits of the action are finite. Since CrystGroups are not finite in general, this is a non-trivial requirement, and so some care is needed.

     Orbit( G, d, opr )
     Orbits( G, D, opr )
     OrbitLengths( G, D, opr )
     Stabilizer( G, d, opr )
     RepresentativeOperation( G, d, e, opr )
     RepresentativesOperation( G, d, opr )

The following functions have a CrystGroup specific implementation, but work for finite CrystGroups only:

     Elements( G )
     ConjugacyClasses( G )
     PermGroup( G )
     SylowSubgroup( G, p )

60.30 Color Groups

Elements of a color group C are colored in the following way. The elements having the same color as C.identity form a subgroup H, which has finite index n in C. H is called the ColorSubgroup of C. Elements of C have the same color if and only if they are in the same right coset of H in C. A fixed list of right cosets of H in C, called ColorCosets, therefore determines a labelling of the colors, which runs from 1 to n. Elements of H by definition have color 1, i.e., the coset with representative C.identity is always the first element of ColorCosets. Right multiplication by a fixed element g of C induces a permutation p(g) of the colors of the parent of C. This defines a natural homomorphism of C into the permutation group of degree n. The image of this homomorphism is called the ColorPermGroup of C, and the homomorphism to it is called the ColorHomomorphism of C.

60.31 ColorGroup

A color group is constructed with

ColorGroup( G, H ),

which returns a colored copy of G, with color subgroup H. G must be a parent group, and H must be a finite index subgroup of G. Color subgroups must be constructed as subgroups of color parent groups, and not by coloring uncolored subgroups. Subgroups of color groups will inherit the coloring of their parent, including the labelling of the colors.

Color groups are identified with a tag isColorGroup. They always have a component colorSubgroup. Color parent groups moreover always have a component colorCosets, which fixes a labelling of the colors.

Groups which may be colored include, in particular, CrystGroups, but coloring of any finite group, such as a finite matrix group or permutation group, should work as well.

60.32 IsColorGroup

IsColorGroup( G )

checks whether G.isColorGroup is bound and true.

60.33 ColorSubgroup

ColorSubgroup( G )

returns the color subgroup of G.

60.34 ColorCosets

ColorCosets( G )

returns the color cosets of G.

60.35 ColorOfElement

ColorOfElement( G, elem )

returns the color of an element.

60.36 ColorPermGroup

ColorPermGroup( G )

returns the ColorPermGroup of G, which is the permutation group induced by G acting on the colors of the parent of G.

60.37 ColorHomomorphism

ColorHomomorphism( G )

returns the homomomorphism from G to its ColorPermGroup.

60.38 Subgroup for color groups

If C is a color group,

Subgroup( C, [elems] )

returns a colored subgroup of C, whereas

C.operations.UncoloredSubgroup( C, [elems] )

returns an ordinary, uncolored subgroup.

60.39 PointGroup for color CrystGroups

If C is a color CrystGroup whose color subgroup is lattice-equal (or translationengleich) with C, the point group of C can consistently be colored. In that case,

PointGroup( C )

returns a colored point group. Otherwise, the point group will be uncolored. An uncolored point group can always be obtained with

C.operations.UncoloredPointGroup( C )

60.40 Inequivalent colorings of space groups

Two colorings of a space group S are equivalent if the two ColorSubgroups are conjugate in the affine normalizer of S.

AffineInequivalentSubgroups( L )

where L is a list of sub space groups with a common parent space group S, returns a list of affine inequivalent subgroups from L. At present, this routine is supported only for CrystGroups constructed from the space group library.

A list of prime index p subgroups of S (actually, a list of conjugacy class representatives of such subgroups) can be obtained with

Filtered( MaximalSubgroupsRepresentatives( S, p ), U - U.index = p )

These two routines together therefore allow to determine all inequivalent colorings of S with p colors.

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GAP 3.4.4
April 1997