Mathieu group M12

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

   12 · 11 · 10 · 9 · 8 = 2⁶ · 3³ · 5 · 11 = 95040.

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) showed that the Schur multiplier of M₁₂ has order 2 (correcting a mistake in (Burgoyne Fong) 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:

• 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.

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	2242	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	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

• Adem, Alejandro; Maginnis, John; Milgram, R. James (1991), "The geometry and cohomology of the Mathieu group M₁₂", Journal of Algebra 139 (1): 90–133, doi:10.1016/0021-8693(91)90285-G, ISSN 0021-8693 
• Burgoyne, N.; Fong, Paul (1966), "The Schur multipliers of the Mathieu groups", Nagoya Mathematical Journal 27 (2): 733–745, doi:10.1017/S0027763000026519, ISSN 0027-7630, http://projecteuclid.org/euclid.nmj/1118801786 
• Burgoyne, N.; Fong, Paul (1968), "A correction to: "The Schur multipliers of the Mathieu groups"", Nagoya Mathematical Journal 31: 297–304, doi:10.1017/S0027763000012782, ISSN 0027-7630, http://projecteuclid.org/euclid.nmj/1118796952 
• Cameron, Peter J. (1999), Permutation Groups, London Mathematical Society Student Texts, 45, Cambridge University Press, ISBN 978-0-521-65378-7, https://archive.org/details/permutationgroup0000came 
• Carmichael, Robert D. (1956), Introduction to the theory of groups of finite order, New York: Dover Publications, ISBN 978-0-486-60300-1, https://books.google.com/books?id=McMgAAAAMAAJ 
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• Conway, John Horton; Parker, Richard A.; Norton, Simon P.; Curtis, R. T.; Wilson, Robert A. (1985), Atlas of finite groups, Oxford University Press, ISBN 978-0-19-853199-9, https://books.google.com/books?id=38fEMl2-Fp8C 
• Conway, John Horton; Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften, 290 (3rd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4757-2016-7, ISBN 978-0-387-98585-5, https://books.google.com/books?id=upYwZ6cQumoC 
• Coxeter, Harold Scott MacDonald (1958), "Twelve points in PG(5,3) with 95040 self-transformations", Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 247 (1250): 279–293, doi:10.1098/rspa.1958.0184, ISSN 0962-8444 
• Curtis, R. T. (1984), "The Steiner system S(5, 6, 12), the Mathieu group M₁₂ and the "kitten"", in Atkinson, Michael D., Computational group theory. Proceedings of the London Mathematical Society symposium held in Durham, July 30–August 9, 1982., Boston, MA: Academic Press, pp. 353–358, ISBN 978-0-12-066270-8, https://books.google.com/books?id=RvvuAAAAMAAJ 
• Cuypers, Hans, The Mathieu groups and their geometries, http://www.win.tue.nl/~hansc/mathieu.pdf 
• Dixon, John D.; Mortimer, Brian (1996), Permutation groups, Graduate Texts in Mathematics, 163, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-0731-3, ISBN 978-0-387-94599-6, https://archive.org/details/permutationgroup0000dixo 
• Frobenius, Ferdinand Georg (1904), "Über die Charaktere der mehrfach transitiven Gruppen" (in German), Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften (Königliche Akademie der Wissenschaften, Berlin) 16: 558–571, Reprinted in volume III of his collected works. 
• Gill, Nick; Hughes, Sam (2019), "The character table of a sharply 5-transitive subgroup of the alternating group of degree 12", International Journal of Group Theory, doi:10.22108/IJGT.2019.115366.1531 
• Griess, Robert L. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-662-03516-0, ISBN 978-3-540-62778-4 
• Hughes, Sam (2018), Representation and Character Theory of the Small Mathieu Groups, https://nickpgill.github.io/MMath_Sam.Hughes.pdf 
• Mathieu, Émile (1861), "Mémoire sur l'étude des fonctions de plusieurs quantités, sur la manière de les former et sur les substitutions qui les laissent invariables", Journal de Mathématiques Pures et Appliquées 6: 241–323, http://gallica.bnf.fr/ark:/12148/bpt6k16405f/f249 
• Thompson, Thomas M. (1983), From error-correcting codes through sphere packings to simple groups, Carus Mathematical Monographs, 21, Mathematical Association of America, ISBN 978-0-88385-023-7, https://books.google.com/books?id=ggqxuG31B3cC 
• Witt, Ernst (1938a), "über Steinersche Systeme", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 12: 265–275, doi:10.1007/BF02948948, ISSN 0025-5858 
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External links

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