## Experimental Mathematics

- Experiment. Math.
- Volume 6, Issue 3 (1997), 247-270.

### The subgroups of {$M\sb {24}$}, or how to compute the table of marks of a finite group

#### 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*. Tables with detailed
information about $M_{24}$ and other groups are included.

#### Article information

**Source**

Experiment. Math., Volume 6, Issue 3 (1997), 247-270.

**Dates**

First available in Project Euclid: 17 March 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1047920424

**Mathematical Reviews number (MathSciNet)**

MR1481593

**Zentralblatt MATH identifier**

0895.20017

**Subjects**

Primary: 20D30: Series and lattices of subgroups

Secondary: 20D08: Simple groups: sporadic groups

**Keywords**

Burnside ring table of marks subgroup lattice Mathieu groups

#### Citation

Pfeiffer, Götz. The subgroups of {$M\sb {24}$}, or how to compute the table of marks of a finite group. Experiment. Math. 6 (1997), no. 3, 247--270. https://projecteuclid.org/euclid.em/1047920424