# ATLAS: Mathieu group M24

Order = 244823040 = 210.33.5.7.11.23.
Mult = 1.
Out = 1.

The following information is available for M24:

### Standard generators

Standard generators of the Mathieu group M24 are a and b where a is in class 2B, b is in class 3A, ab has order 23 and abababbababbabb has order 4.

### Black box algorithms

#### Finding generators

To find standard generators for M24:
• Find any element of order 10. Its fifth power is a 2B-element, x, say.
• Find any element of order 15. Its fifth power is a 3A-element, y, say.
• Find a conjugate a of x and a conjugate b of y such that ab has order 23.
• If ab(ababb)2abb has order 5, replace b by its inverse.
This algorithm is available in computer readable format: finder for M24.

#### Checking generators

To check that elements x and y of M24 are standard generators:
• Check o(x) = 2
• Check o(y) = 3
• Check o(xy) = 23
• Check o(xyxyxyyxyxyyxyy) = 4
• Check o(xyxyxyy) = 12
This algorithm is available in computer readable format: checker for M24.

### Presentation

A presentation of M24 on its standard generators is given below.

< a, b | a2 = b3 = (ab)23 = [a, b]12 = [a, bab]5 = (ababab-1)3(abab-1ab-1)3 = (ab(abab-1)3)4 = 1 >.

This presentation is available in Magma format as follows: M24 on a and b.

### Representations

The representations of M24 available are:
• Some primitive permutation representations
• All 2-modular irreducible representations.
• Some 3-modular irreducible representations.
• Some 5-modular irreducible representations.
• Dimension 23 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 45 over GF(25): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- courtesy of Stephen Rogers.
• Dimension 231 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 252 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 253 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 770a over GF(25): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 990 over GF(25): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some 7-modular irreducible representations.
• Some 11-modular irreducible representations.
• Dimension 23 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 45 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- courtesy of Stephen Rogers.
• Dimension 229 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 231 over GF(121): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 253 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 482 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 770a over GF(121): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 806 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 990b over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some 23-modular irreducible representations.
• a and b as 23 × 23 matrices over Z - see Version 1.

### Maximal subgroups

The maximal subgroups of M24 are as follows. Words provided by Peter Walsh, implemented and checked by Ibrahim Suleiman.

### Conjugacy classes

A set of generators for the maximal cyclic subgroups can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements. Problems of algebraic conjugacy are not yet dealt with.
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