Mathieu group M23

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In mathematics, the Mathieu group M23, introduced by Mathieu (1861, 1873), is a 4-fold transitive permutation group on 23 objects of order

27 · 32 ··· 11 · 23 (= 10200960).

Properties[edit]

The Schur multiplier and the outer automorphism group are both trivial.

Milgram (2000) calculated the integral cohomology, and showed in particular that M23 has the unusual property that the first 4 integral homology groups all vanish

Representations[edit]

M23 is the point stabilizer of the action of the Mathieu group M24 on 24 points, giving it a 4-transitive permutation representation on 23 points with point stabilizer the Mathieu group M22.

M23 has 2 different rank 3 actions on 253 points. One is the action on unordered pairs with orbit sizes 1+42+210 and point stabilizer M2.2, and the other is the action on heptads with orbit sizes 1+112+140 and point stabilizer 24.A7.

The integral representation corresponding to the permutation action on 23 points decomposes into the trivial representation and a 22-dimensional representation. The 23 dimensional representation gives an irreducible representation over any field of characteristic not 2 or 23.

Over the field of order 2, it has 2 11-dimensional representations, the restrictions of the corresponding representations of the Mathieu group M24.

Maximal subgroups[edit]

  • M22, order 443520
  • PSL(3,4):2, order 40320, orbits of 21 and 2
  • 24:A7, order 40320, orbits of 7 and 16
Stabilizer of W23 block
  • A8, order 20160, orbits of 8 and 15
  • M11, order 7920, orbits of 11 and 12
  • (24:A5):S3 or M20:S3, order 5760, orbits of 3 and 20 (5 blocks of 4)
One-point stabilizer of the sextet group
  • 23:11, order 253, simply transitive


Conjugacy classes[edit]

Order No. elements Cycle structure
1 = 1 1 123
2 = 2 3795 = 3 · 5 · 11 · 23 1728
3 = 3 56672 = 25 · 7 · 11 · 23 1536
4 = 22 318780 = 22 · 32 · 5 · 7 · 11 · 23 132244
5 = 5 680064 = 27 · 3 · 7 · 11 · 23 1354
6 = 2 · 3 850080 = 25 · 3 · 5 · 7 · 11 · 23 1·223262
7 = 7 728640 = 26 · 32 · 5 · 11 · 23 1273
7 = 7 728640 = 26 · 32 · 5 · 11 · 23 1273
8 = 23 1275120 = 24 · 32 · 5 · 7 · 11 · 23 1·2·4·82
11 = 11 927360= 27 · 32 · 5 · 7 · 23 1·112
11 = 11 927360= 27 · 32 · 5 · 7 · 23 1·112
14 = 2 · 7 728640= 26 · 32 · 5 · 11 · 23 2·7·14
14 = 2 · 7 728640= 26 · 32 · 5 · 11 · 23 2·7·14
15 = 3 · 5 680064= 27 · 3 · 7 · 11 · 23 3·5·15
15 = 3 · 5 680064= 27 · 3 · 7 · 11 · 23 3·5·15
23 = 23 443520= 27 · 32 · 5 · 7 · 11 23
23 = 23 443520= 27 · 32 · 5 · 7 · 11 23

References[edit]

External links[edit]