Mathieu group M12

M₁₂ is one of the 26 sporadic groups and was introduced by {{harvs|txt |authorlink=Émile Léonard Mathieu |last=Mathieu |year1=1861 |year2=1873}}. It is a sharply 5-transitive permutation group on 12 objects. {{harvtxt|Burgoyne|Fong|1968}} showed that the Schur multiplier of M₁₂ has order 2 (correcting a mistake in {{harv|Burgoyne|Fong|1966}} where they incorrectly claimed it has order 1).

The double cover had been implicitly found earlier by {{harvtxt|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.

{{harvtxt|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:

class="wikitable" |+ Maximal subgroups of M12
No.	Structure	Order	Index	Comments
1,2	M11	align=right|7,920 = 24·32·5·11	align=right| 12 = 22·3	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	S6:2 ≅ M10:2	align=right|1,440 = 25·32·5	align=right| 66 = 2·3·11	two classes, exchanged by an outer automorphism. The outer automorphism group of the symmetric group S6. 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	L2(11)	align=right| 660 = 22·3·5·11	align=right| 144 = 24·32	doubly transitive on the 12 points
6,7	32:(2.S4)	align=right| 432 = 24·33	align=right| 220 = 22·5·11	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 C3 x C3.
8	S5 x 2	align=right| 240 = 24·3·5	align=right| 396 = 22·32·11	doubly imprimitive on 6 sets of 2 points; centralizer of a sextuple transposition
9	Q8:S4	align=right| 192 = 26·3	align=right| 495 = 32·5·11	orbits of sizes 4 and 8; centralizer of a quadruple transposition (an involution of class 2B)
10	42:(2 x S3)	align=right| 192 = 26·3	align=right| 495 = 32·5·11	imprimitive on 3 sets of size 4
11	A4 x S3	align=right| 72 = 23·32	align=right|1,320 = 23·3·5·11	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.

class="wikitable" style="text-align: right"
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	2242	rowspan="2"|Fused 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	rowspan="2"|Fused under an outer automorphism
8	11880	8	4 8
10	9504	10	2 10
11	8640	11	1 11	rowspan="2"|Fused under an outer automorphism
11	8640	11	1 11

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• [http://mathworld.wolfram.com/MathieuGroups.html MathWorld: Mathieu Groups]
• [http://brauer.maths.qmul.ac.uk/Atlas/v3/group/M12/ Atlas of Finite Group Representations: M₁₂]

{{DEFAULTSORT:Mathieu Group M12}}

Category:Sporadic groups