Götz Pfeiffer,

The Subgroups of M₂₄ 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.

The Subgroups of M24 or How to Compute the Table of Marks of a Finite Group.

Abstract.

The Subgroups of M₂₄ or How to Compute the Table of Marks of a Finite Group.