In the area of modern algebra known as group theory, the **Mathieu group** *M*_{23} is a sporadic simple group of order

2

^{7} **·** 3

^{2} **·** 5

**·** 7

**·** 11

**·** 23 = 10200960
≈ 1

×10

^{7}.

*M*_{23} is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 4-fold transitive permutation group on 23 objects. The Schur multiplier and the outer automorphism group are both trivial.

Milgram (2000) calculated the integral cohomology, and showed in particular that M_{23} has the unusual property that the first 4 integral homology groups all vanish.

The inverse Galois problem seems to be unsolved for M_{23}. In other words, no polynomial in Z[x] seems to be known to have M_{23} as its Galois group. The inverse Galois problem is solved for all other sporadic simple groups.

M_{23} is the point stabilizer of the action of the Mathieu group M24 on 24 points, giving it a 4-transitive permutation representation on 23 points with point stabilizer the Mathieu group M22.

M_{23} has 2 different rank 3 actions on 253 points. One is the action on unordered pairs with orbit sizes 1+42+210 and point stabilizer M_{21}.2, and the other is the action on heptads with orbit sizes 1+112+140 and point stabilizer 2^{4}.A_{7}.

The integral representation corresponding to the permutation action on 23 points decomposes into the trivial representation and a 22-dimensional representation. The 23 dimensional representation gives an irreducible representation over any field of characteristic not 2 or 23.

Over the field of order 2, it has 2 11-dimensional representations, the restrictions of the corresponding representations of the Mathieu group M24.

There are 7 conjugacy classes of maximal subgroups of *M*_{23} as follows:

M_{22}, order 443520
PSL(3,4):2, order 40320, orbits of 21 and 2
2^{4}:A_{7}, order 40320, orbits of 7 and 16
Stabilizer of W

_{23} block

A_{8}, order 20160, orbits of 8 and 15
M_{11}, order 7920, orbits of 11 and 12
(2^{4}:A_{5}):S_{3} or M_{20}:S_{3}, order 5760, orbits of 3 and 20 (5 blocks of 4)
One-point stabilizer of the sextet group

23:11, order 253, simply transitive