Mathieu group M12

For general background and history of the Mathieu sporadic groups, see Mathieu group.

In the area of modern algebra known as group theory, the Mathieu group M12 is a sporadic simple group of order

   26 · 33 · 5 · 11 = 95040 = 12×11×10×9×8
≈ 1×105.

History and properties

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.

Representations

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.

Maximal subgroups

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

Isomorphic to the affine group on the space C3 x C3.
Centralizer of a sextuple transposition
Centralizer of a quadruple transposition

Conjugacy classes

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.

Order Number Centralizer Cycles Fusion
1 1 95040 112
2 396 240 26
2 495 192 1424
3 1760 54 1333
3 2640 36 34
4 2970 32 2242Fused under an outer automorphism
4 2970 32 1442
5 9504 10 1252
6 7920 12 62
6 15840 6 1 2 3 6
8 11880 8 122 8 Fused under an outer automorphism
8 11880 8 4 8
10 9504 10 2 10
11 8640 11 1 11 Fused under an outer automorphism
11 8640 11 1 11

References

External links

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