M₂₃ 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 M₂₃ has the unusual property that the first 4 integral homology groups all vanish.

The inverse Galois problem seems to be unsolved for M₂₃. In other words, no polynomial in Z[x] seems to be known to have M₂₃ as its Galois group. The inverse Galois problem is solved for all other sporadic simple groups.

Construction using finite fields edit

Let F_2¹¹ be the finite field with 2¹¹ elements. Its group of units has order 2¹¹ − 1 = 2047 = 23 · 89, so it has a cyclic subgroup C of order 23.

The Mathieu group M₂₃ can be identified with the group of F₂-linear automorphisms of F_2¹¹ that stabilize C. More precisely, the action of this automorphism group on C can be identified with the 4-fold transitive action of M₂₃ on 23 objects.

M₂₃ 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.

M₂₃ 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 M₂₁.2, and the other is the action on heptads with orbit sizes 1+112+140 and point stabilizer 2⁴.A₇.

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

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

There are 7 conjugacy classes of maximal subgroups of M₂₃ as follows:

• M₂₂, order 443520
• PSL(3,4):2, order 40320, orbits of 21 and 2
• 2⁴:A₇, order 40320, orbits of 7 and 16

Stabilizer of W₂₃ block

• A₈, order 20160, orbits of 8 and 15
• M₁₁, order 7920, orbits of 11 and 12
• (2⁴:A₅):S₃ or M₂₀:S₃, order 5760, orbits of 3 and 20 (5 blocks of 4)

One-point stabilizer of the sextet group

• 23:11, order 253, simply transitive

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• MathWorld: Mathieu Groups
• Atlas of Finite Group Representations: M₂₃