Götz Pfeiffer,
# The Subgroups of M_{24} or
How to Compute the Table of Marks of a Finite Group.

*Experiment. Math.* **6** (1997), 247-270.

## Abstract.

Let $G$ be a finite group. The table of marks of $G$ arises from a
characterization of the permutation representations of $G$ by certain
numbers of fixed points. It provides a compact description of the subgroup
lattice of $G$ and enables explicit calculations in the Burnside ring of
$G$. In this article we introduce a method for constructing the table of
marks of $G$ from tables of marks of proper subgroups of $G$. An
implementation of this method is available in the GAP language. These
computer programs are used to construct the
table of marks of the sporadic
simple Mathieu group $M$_{24}. The final section describes how to derive
information about the structure of $G$ from its table of marks via the
investigation of certain Möbius functions and the idempotents of the
Burnside ring of $G$. The appendix contains tables with detailed
information about $M$_{24} and other groups.
Available as
DVI (147 kB) and as gzip'ed
PostScript file (164 kB).

The actual table of all 1529 conjugacy classes
of subgroups of $M$_{24} is available as a separate document.