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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:

• 
  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.

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, MR 1106342.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
  


• 
  Burgoyne, N.; Fong, Paul (1966), "The Schur multipliers of the Mathieu groups", Nagoya Mathematical Journal, 27: 733–745, ISSN 0027-7630, MR 0197542
  


• 
  Burgoyne, N.; Fong, Paul (1968), "A correction to: "The Schur multipliers of the Mathieu groups"", Nagoya Mathematical Journal, 31: 297–304, ISSN 0027-7630, MR 0219626
  


• 
  Cameron, Peter J. (1999), Permutation Groups, London Mathematical Society Student Texts, 45, Cambridge University Press, ISBN 978-0-521-65378-7
  


• 
  Carmichael, Robert D. (1956) [1937], Introduction to the theory of groups of finite order, New York: Dover Publications, ISBN 978-0-486-60300-1, MR 0075938
  


• 
  Conway, John Horton (1971), "Three lectures on exceptional groups", in Powell, M. B.; Higman, Graham, Finite simple groups, Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969., Boston, MA: Academic Press, pp. 215–247, ISBN 978-0-12-563850-0, MR 0338152 Reprinted in Conway & Sloane (1999, 267–298)


• 
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• 
  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, MR 0920369
  


• 
  Coxeter, Harold Scott MacDonald (1958), "Twelve points in PG(5,3) with 95040 self-transformations", Proceedings of the Royal Society. London. Series A. Mathematical, Physical and Engineering Sciences, 247: 279–293, doi:10.1098/rspa.1958.0184, ISSN 0962-8444, JSTOR 100667, MR 0120289
  


• 
  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, MR 0760669
  


• 
  Cuypers, Hans, The Mathieu groups and their geometries (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, MR 1409812
  


• 
  Frobenius, Ferdinand Georg (1904), "Über die Charaktere der mehrfach transitiven Gruppen", Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften (in German), Königliche Akademie der Wissenschaften, Berlin, 16: 558–571, Reprinted in volume III of his collected works.
  


• 
  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, MR 1707296
  


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  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
  


• 
  Mathieu, Émile (1873), "Sur la fonction cinq fois transitive de 24 quantités", Journal de Mathématiques Pures et Appliquées (in French), 18: 25–46, JFM 05.0088.01
  


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  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, MR 0749038
  


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• 
  Witt, Ernst (1938b), "Die 5-fach transitiven Gruppen von Mathieu", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 12: 256–264, doi:10.1007/BF02948947
  


• 
  Hughes, Sam (2018), Representation and Character Theory of the Small Mathieu Groups (PDF)
  

External links

• MathWorld: Mathieu Groups


• Atlas of Finite Group Representations: M₁₂