Mathieu group M₁₂

M₁₂ is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a sharply 5-transitive permutation group on 12 objects. Burgoyne & Fong (1968) showed that the Schur multiplier of M₁₂ has order 2 (correcting a mistake in (Burgoyne & Fong 1966) where they incorrectly claimed it has order 1).

The double cover had been implicitly found earlier by Coxeter (1958), who showed that M₁₂ is a subgroup of the projective linear group of dimension 6 over the finite field with 3 elements.

The outer automorphism group has order 2, and the full automorphism group M₁₂.2 is contained in M₂₄ as the stabilizer of a pair of complementary dodecads of 24 points, with outer automorphisms of M₁₂ swapping the two dodecads.

Frobenius (1904) calculated the complex character table of M₁₂.

M₁₂ has a strictly 5-transitive permutation representation on 12 points, whose point stabilizer is the Mathieu group M₁₁. Identifying the 12 points with the projective line over the field of 11 elements, M₁₂ is generated by the permutations of PSL₂(11) together with the permutation (2,10)(3,4)(5,9)(6,7). This permutation representation preserves a Steiner system S(5,6,12) of 132 special hexads, such that each pentad is contained in exactly 1 special hexad, and the hexads are the supports of the weight 6 codewords of the extended ternary Golay code. In fact M₁₂ has two inequivalent actions on 12 points, exchanged by an outer automorphism; these are analogous to the two inequivalent actions of the symmetric group S₆ on 6 points.

The double cover 2.M₁₂ is the automorphism group of the extended ternary Golay code, a dimension 6 length 12 code over the field of order 3 of minimum weight 6. In particular the double cover has an irreducible 6-dimensional representation over the field of 3 elements.

The double cover 2.M₁₂ is the automorphism group of any 12×12 Hadamard matrix.

M₁₂ centralizes an element of order 11 in the monster group, as a result of which it acts naturally on a vertex algebra over the field with 11 elements, given as the Tate cohomology of the monster vertex algebra.

There are 11 conjugacy classes of maximal subgroups of M₁₂, 6 occurring in automorphic pairs, as follows:

• M₁₁, order 7920, index 12. There are two classes of maximal subgroups, exchanged by an outer automorphism. One is the subgroup fixing a point with orbits of size 1 and 11, while the other acts transitively on 12 points.
• S₆:2 = M₁₀.2, the outer automorphism group of the symmetric group S₆ of order 1440, index 66. There are two classes of maximal subgroups, exchanged by an outer automorphism. One is imprimitive and transitive, acting with 2 blocks of 6, while the other is the subgroup fixing a pair of points and has orbits of size 2 and 10.
• PSL(2,11), order 660, index 144, doubly transitive on the 12 points
• 3²:(2.S₄), order 432. There are two classes of maximal subgroups, exchanged by an outer automorphism. One acts with orbits of 3 and 9, and the other is imprimitive on 4 sets of 3.

Isomorphic to the affine group on the space C₃ x C₃.

• S₅ x 2, order 240, doubly imprimitive on 6 sets of 2 points

Centralizer of a sextuple transposition

• Q:S₄, order 192, orbits of 4 and 8.

Centralizer of a quadruple transposition

• 4²:(2 x S₃), order 192, imprimitive on 3 sets of 4
• A₄ x S₃, order 72, doubly imprimitive, 4 sets of 3 points.

The cycle shape of an element and its conjugate under an outer automorphism are related in the following way: the union of the two cycle shapes is balanced, in other words invariant under changing each n-cycle to an N/n cycle for some integer N.

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• MathWorld: Mathieu Groups
• Atlas of Finite Group Representations: M₁₂