ATLAS: Mathieu group M_{24}
Order = 244823040 = 2^{10}.3^{3}.5.7.11.23.
Mult = 1.
Out = 1.
The following information is available for M_{24}:
Standard generators of the Mathieu group M_{24} are a and
b where a is in class 2B, b is in class 3A,
ab has order 23 and abababbababbabb has order 4.
Finding generators
To find standard generators for M_{24}:
 Find any element of order 10. Its fifth power is a 2Belement, x, say.
 Find any element of order 15. Its fifth power is a 3Aelement, y, say.
 Find a conjugate a of x and a conjugate b of y
such that ab has order 23.
 If ab(ababb)^{2}abb has order 5, replace b
by its inverse.
This algorithm is available in computer readable format:
finder for M_{24}.
Checking generators
To check that elements x and y of M_{24}
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 M_{24}.
A presentation of M_{24} on its standard generators is given below.
< a, b  a^{2} = b^{3} = (ab)^{23} = [a, b]^{12} = [a, bab]^{5} = (ababab^{1})^{3}(abab^{1}ab^{1})^{3} = (ab(abab^{1})^{3})^{4} = 1 >.
This presentation is available in Magma format as follows:
M24 on a and b.
The representations of M_{24} available are:
 Some primitive permutation representations

Permutations on 24 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 276 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 759 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 1288 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 1771 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 2024 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 3795 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All 2modular irreducible representations.

Dimension 11 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the Golay code.

Dimension 11 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the Golay cocode.

Dimension 44 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 44 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 120 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 220 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 220 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 252 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 320 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 320 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 1242 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 1792 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some 3modular irreducible representations.

Dimension 22 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 45 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 courtesy of Stephen Rogers.

Dimension 231 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 252 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 483 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 770 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 770 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 990 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some 5modular 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 7modular irreducible representations.

Dimension 23 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 45 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 courtesy of Stephen Rogers.

Dimension 231 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 252 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 253 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 483 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 770 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 990 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some 11modular 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 23modular irreducible representations.

Dimension 23 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 45 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 231 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 251 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 253 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 483 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 770 over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 990b over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 a and b as
23 × 23 matrices over Z  see Version 1.
The maximal subgroups of M_{24} are as follows. Words provided by
Peter Walsh, implemented and checked by Ibrahim Suleiman.
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.
Go to main ATLAS (version 2.0) page.
Go to sporadic groups page.
Go to old M24 page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 7th June 2000.
Last updated 21.12.04 by SJN.
Information checked to
Level 0 on 07.06.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.