In the area of modern algebra known as group theory, the Mathieu group M<sub>12</sub> is a sporadic simple group of order
M<sub>12</sub> is one of the 26 sporadic groups and was introduced by . It is a sharply 5-transitive permutation group on 12 objects. showed that the Schur multiplier of M<sub>12</sub> has order 2 (correcting a mistake in where they incorrectly claimed it has order 1).
The double cover had been implicitly found earlier by , who showed that M<sub>12</sub> 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 M<sub>12</sub>.2 is contained in M<sub>24</sub> as the stabilizer of a pair of complementary dodecads of 24 points, with outer automorphisms of M<sub>12</sub> swapping the two dodecads.
calculated the complex character table of M<sub>12</sub>.
M<sub>12</sub> has a strictly 5-transitive permutation representation on 12 points, whose point stabilizer is the Mathieu group M<sub>11</sub>. Identifying the 12 points with the projective line over the field of 11 elements, M<sub>12</sub> is generated by the permutations of PSL<sub>2</sub>(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 M<sub>12</sub> has two inequivalent actions on 12 points, exchanged by an outer automorphism; these are analogous to the two inequivalent actions of the symmetric group S<sub>6</sub> on 6 points.
The double cover 2.M<sub>12</sub> 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.M<sub>12</sub> is the automorphism group of any 12ÃÂ12 Hadamard matrix.
M<sub>12</sub> 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 M<sub>12</sub>, 6 occurring in automorphic pairs, as follows:
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.