# Mathieu group M12

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In the area of modern algebra known as group theory, the Mathieu group M$_{12}$ is a sporadic simple group of order
    95,040 = 12 · 11 · 10 · 9 · 8 = 2$^6$ · 3$^3$ · 5 · 11.

## History and properties

M$_{12}$ 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 M$_{12}$ 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 M$_{12}$ 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$_{12}$.2 is contained in M$_{24}$ as the stabilizer of a pair of complementary dodecads of 24 points, with outer automorphisms of M$_{12}$ swapping the two dodecads.

## Representations

Frobenius (1904) calculated the complex character table of M$_{12}$.
M$_{12}$ has a strictly 5-transitive permutation representation on 12 points, whose point stabilizer is the Mathieu group M$_{11}$. Identifying the 12 points with the projective line over the field of 11 elements, M$_{12}$ is generated by the permutations of PSL$_2$(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$_{12}$ 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$_6$ on 6 points.
The double cover 2.M$_{12}$ 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$_{12}$ is the automorphism group of any 12×12 Hadamard matrix.
M$_{12}$ 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 M$_{12}$, 6 occurring in automorphic pairs, as follows:
Maximal subgroups of M$_{12}$ :
| No. | Structure | Order | Index | Comments |
| 1,2 | M$_{11}$ | 7,920 = 2$^4$·3$^2$·5·11 | 12 = 2$^2$·3 | two classes, exchanged by an outer automorphism. One is the subgroup fixing a point with orbits of sizes 1 and 11, while the other acts transitively on the 12 points. |
| 3,4 | S$_6$:2 ≅ M$_{10}$:2 | 1,440 = 2$^5$·3$^2$·5 | 66 = 2·3·11 | two classes, exchanged by an outer automorphism. The outer automorphism group of the symmetric group S$_6$. One class is imprimitive and transitive, acting with 2 blocks of size 6, while the other is the subgroup fixing a pair of points and has orbits of sizes 2 and 10. |
| 5 | L$_2$(11) | 660 = 2$^2$·3·5·11 | 144 = 2$^4$·3$^2$ | doubly transitive on the 12 points |
| 6,7 | 3$^2$:(2.S$_4$) | 432 = 2$^4$·3$^3$ | 220 = 2$^2$·5·11 | two classes, exchanged by an outer automorphism. One acts with orbits of sizes 3 and 9, and the other is imprimitive on 4 sets of size 3; isomorphic to the affine group on the space C$_3$ x C$_3$. |
| 8 | S$_5$ x 2 | 240 = 2$^4$·3·5 | 396 = 2$^2$·3$^2$·11 | doubly imprimitive on 6 sets of 2 points; centralizer of a sextuple transposition |
| 9 | Q$_8$:S$_4$ | 192 = 2$^6$·3 | 495 = 3$^2$·5·11 | orbits of sizes 4 and 8; centralizer of a quadruple transposition (an involution of class 2B) |
| 10 | 4$^2$:(2 x S$_3$) | 192 = 2$^6$·3 | 495 = 3$^2$·5·11 | imprimitive on 3 sets of size 4 |
| 11 | A$_4$ x S$_3$ | 72 = 2$^3$·3$^2$ | 1,320 = 2$^3$·3·5·11 | doubly imprimitive, 4 sets of 3 points |

## 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 | 1$^{12}$ |
| 2 | 396 | 240 | 2$^6$ |
| 2 | 495 | 192 | 1$^4$2$^4$ |
| 3 | 1760 | 54 | 1$^3$3$^3$ |
| 3 | 2640 | 36 | 3$^4$ |
| 4 | 2970 | 32 | 2$^2$4$^2$ | Fused under an outer automorphism |
| 4 | 2970 | 32 | 1$^4$4$^2$ | Fused under an outer automorphism |
| 5 | 9504 | 10 | 1$^2$5$^2$ |
| 6 | 7920 | 12 | 6$^2$ |
| 6 | 15840 | 6 | 1 2 3 6 |
| 8 | 11880 | 8 | 1$^2$2 8 | Fused under an outer automorphism |
| 8 | 11880 | 8 | 4 8 | Fused under an outer automorphism |
| 10 | 9504 | 10 | 2 10 |
| 11 | 8640 | 11 | 1 11 | Fused under an outer automorphism |
| 11 | 8640 | 11 | 1 11 | Fused under an outer automorphism |