ATLAS: Mathieu group M_{24}
Order = 244823040 = 2^{10}.3^{3}.5.7.11.23.
Mult = 1.
Out = 1.
Standard generators
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.
Black box algorithms
To find standard generators for M_{24}:
- 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)^{2}abb has order 4.
Presentation
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 >.
Representations
The representations of M_{24} 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 M_{24} 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.
Return to main ATLAS page.
Last updated 21st December 1999,
R.A.Wilson, R.A.Parker and J.N.Bray