# Mathieu group M11

In mathematics, the Mathieu group M11, introduced by Mathieu (1861, 1873), is a sharply 4-transitive permutation group on 11 objects, of order

24 · 32 ·· 11 (= 7920).

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

## Construction

The Mathieu group can be defined as a permutation group on 11 points generated by some set of permutations, such as the pair (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) of permutations used by the GAP computer algebra system.

## Representations

M11 has a strictly 4-transitive permutation representation on 11 points, whose point stabilizer is sometimes denoted by M10, and is a non-split extension of the form A6.2 (an extension of the group of order 2 by the alternating group A6). This action is the automorphism group of a Steiner system S(4,5,11). The induced action on unordered pairs of points gives a rank 3 action on 55 points.

M11 has a 2-transitive permutation representation on 12 points with point stabilizer PSL2(11). The permutation representations on 11 and 12 points can both be seen inside the Mathieu group M12 as two different embeddings of M11 in M12, exchanged by an outer automorphism.

The permutation representation on 11 points gives a complex irreducible representation in 10 dimensions. This is the smallest possible dimension of a faithful complex representation, though there are also two other such representations in 10 dimensions forming a complex conjugate pair.

M11 has two 5-dimensional irreducible representations over the field with 5 elements, related to the restrictions of 6-dimensional representations of the double cover of M12. These have the smallest dimension of any faithful linear representations of M11 over any field.

## Maximal subgroups

There are 5 conjugacy classes of maximal subgroups.

• M10, order 720, one-point stabilizer in representation of degree 11
• PSL(2,11), order 660, one-point stabilizer in representation of degree 12
• M9:2, order 144, stabilizer of a 9 and 2 partition.
• S5, order 120, orbits of 5 and 6
Stabilizer of block in the S(4,5,11) Steiner system
• Q:S3, order 48, orbits of 8 and 3
Centralizer of a quadruple transposition
Isomorphic to GL(2,3).

## Number of elements of each order

The maximum order of any element in M11 is 11. The conjugacy class orders and sizes are found in the ATLAS.ATLAS: Mathieu group M11

Order No. elements
1 = 1 1 = 1
2 = 2 165 = 3 · 5 · 11
3 = 3 440 = 23 · 5 · 11
4 = 22 990 = 2 · 32 · 5 · 11
5 = 5 1584 = 24 · 32 · 11
6 = 2 · 3 1320 = 23 · 3 · 5 · 11
8 = 23 990 = 2 · 32 · 5 · 11
8 = 23 990 = 2 · 32 · 5 · 11
11 = 11 720 = 24 · 32 · 5
11 = 11 720 = 24 · 32 · 5