# ATLAS: Mathieu group M24

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

### 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

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 and ab(ababb)2abb has order 4.

### 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 >.

### Representations

The representations of M24 available are:
• Some primitive permutation representations
• a and b as permutations on 24 points.
• a and b as permutations on 276 points.
• a and b as permutations on 759 points.
• a and b as permutations on 1288 points.
• a and b as permutations on 1771 points.
• a and b as permutations on 2024 points.
• a and b as permutations on 3795 points.
• All 2-modular irreducible representations.
• a and b as 11 × 11 matrices over GF(2) - the Golay code.
• a and b as 11 × 11 matrices over GF(2) - the Golay cocode.
• a and b as 44 × 44 matrices over GF(2).
• a and b as 44 × 44 matrices over GF(2) - the dual of the above.
• a and b as 120 × 120 matrices over GF(2).
• a and b as 220 × 220 matrices over GF(2).
• a and b as 220 × 220 matrices over GF(2) - the dual of the above.
• a and b as 252 × 252 matrices over GF(2).
• a and b as 320 × 320 matrices over GF(2).
• a and b as 320 × 320 matrices over GF(2) - the dual of the above.
• a and b as 1242 × 1242 matrices over GF(2).
• a and b as 1792 × 1792 matrices over GF(2).
• a and b as 22 × 22 matrices over GF(3).
• a and b as 231 × 231 matrices over GF(3).
• a and b as 252 × 252 matrices over GF(3).
• a and b as 23 × 23 matrices over GF(5).
• a and b as 23 × 23 matrices over GF(7).
• a and b as 45 × 45 matrices over GF(7) - courtesy of Stephen Rogers.
• a and b as 23 × 23 matrices over GF(11).
• a and b as 45 × 45 matrices over GF(11) - courtesy of Stephen Rogers.
• a and b as 23 × 23 matrices over GF(23).
• a and b as 45 × 45 matrices over GF(23).
• a and b as 23 × 23 matrices over Z.

### 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.