# ATLAS: Mathieu group M23

Order = 10200960.
Mult = 1.
Out = 1.

### Standard generators

Standard generators of the Mathieu group M23 are a and b where a has order 2, b has order 4, ab has order 23, and ababababbababbabb has order 8.

### Black box algorithms

To find standard generators for M23:
• Find any elements x of order 2 and y of order 4.
• Find a conjugate a of x and a conjugate b of y, whose product has order 23, such that (ab)^2(ababb)^2abb has order 8 or 11. In the latter case, replace b by its inverse.

### Representations

The representations available are as follows. They should be in Atlas order, defined by setting ab in 23B, abababb in 15A, abababbb in 7A and ababababb in 11B.
• Faithful irreducibles in characteristic 2.
• a and b as 11 x 11 matrices over GF(2) - the co-code representation.
• a and b as 11 x 11 matrices over GF(2) - the code representation.
• a and b as 44 x 44 matrices over GF(2).
• a and b as 44 x 44 matrices over GF(2) - the dual of the above.
• a and b as 120 x 120 matrices over GF(2).
• a and b as 220 x 220 matrices over GF(2).
• a and b as 220 x 220 matrices over GF(2) - the dual of the above.
• a and b as 252 x 252 matrices over GF(2).
• Faithful irreducibles in characteristic 3.
• a and b as 22 x 22 matrices over GF(3).
• a and b as 104 x 104 matrices over GF(3) - phi5 in the modular atlas.
• a and b as 104 x 104 matrices over GF(3) - phi6 in the modular atlas.
• a and b as 231 x 231 matrices over GF(3).
• a and b as 22 x 22 matrices over GF(5).
• Faithful irreducibles in characteristic 7.
• a and b as 22 x 22 matrices over GF(7).
• a and b as 45 x 45 matrices over GF(7).
• a and b as 208 x 208 matrices over GF(7).
• a and b as 231 x 231 matrices over GF(7).
• a and b as 22 x 22 matrices over GF(11).
• a and b as 45 x 45 matrices over GF(11).
• a and b as 21 x 21 matrices over GF(23).
• a and b as 45 x 45 matrices over GF(23).
• a and b as 280 x 280 matrices over GF(23).
• Primitive permutation representations.
• a and b as permutations on 23 points.
• a and b as permutations on 253 points - the cosets of L3(4).2.
• a and b as permutations on 253 points - the cosets of 2^4.A7.
• a and b as permutations on 506 points.
• a and b as permutations on 1288 points.
• a and b as permutations on 1771 points.

### Maximal subgroups

The maximal subgroups of M23 are as follows.

### 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.
Last updated 23.03.99

R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk