# Mathieu group M11

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vte :
In the area of modern algebra known as group theory, the Mathieu group M$_{11}$ is a sporadic simple group of order
    7,920 = 11 · 10 · 9 · 8 = 2$^4$ · 3$^2$ · 5 · 11.

## History and properties

M$_{11}$ is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is the smallest sporadic group and, along with the other four Mathieu groups, the first to be discovered. The Schur multiplier and the outer automorphism group are both trivial.
M$_{11}$ is a sharply 4-transitive permutation group on 11 objects. It admits many generating sets 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

M$_{11}$ has a sharply 4-transitive permutation representation on 11 points. The point stabilizer is sometimes denoted by M$_{10}$, and is a non-split extension of the form A$_6$.2 (an extension of the group of order 2 by the alternating group A$_6$). 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.
M$_{11}$ has a 3-transitive permutation representation on 12 points with point stabilizer PSL$_2$(11). The permutation representations on 11 and 12 points can both be seen inside the Mathieu group M$_{12}$ as two different embeddings of M$_{11}$ in M$_{12}$, 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.
M$_{11}$ has two 5-dimensional irreducible representations over the field with 3 elements, related to the restrictions of 6-dimensional representations of the double cover of M$_{12}$. These have the smallest dimension of any faithful linear representations of M$_{11}$ over any field.

## Maximal subgroups

There are 5 conjugacy classes of maximal subgroups of M$_{11}$ as follows:
Maximal subgroups of M$_{11}$ :
| No. | Structure | Order | Index | Comments |
| 1 | M$_{10}$ ≅ A$_6$$^·$2 | 720 = 2$^4$·3$^2$·5 | 11 | one-point stabilizer in representation of degree 11 |
| 2 | L$_2$(11) | 660 = 2$^2$·3·5·11 | 12 = 2$^2$·3 | one-point stabilizer in representation of degree 12 |
| 3 | M$_9$:2 ≅ 3$_2$:Q$_8$.2 | 144 = 2$^4$·3$^2$ | 55 = 5·11 | stabilizer of a 9 and 2 partition |
| 4 | S$_5$ | 120 = 2$^3$·3·5 | 66 = 2·3·11 | has orbits of sizes 5 and 6; stabilizer of block in the S(4,5,11) Steiner system |
| 5 | Q$_8$:S$_3$ ≅ GL$_2$(3) | 48 = 2$^4$·3 | 165 = 3·5·11 | has orbits of sizes 8 and 3; centralizer of a quadruple transposition |

## Conjugacy classes

The maximum order of any element in M$_{11}$ is 11. Cycle structures are shown for the representations both of degree 11 and 12.
| Order | No. elements | Degree 11 | Degree 12 | |
| 1 = 1 | 1 = 1 | 1$^{11}$· | 1$^{12}$· |
| 2 = 2 | 165 = 3 · 5 · 11 | 1$^3$·2$^4$ | 1$^4$·2$^4$ |
| 3 = 3 | 440 = 2$^3$ · 5 · 11 | 1$^2$·3$^3$ | 1$^3$·3$^3$ |
| 4 = 2$^2$ | 990 = 2 · 3$^2$ · 5 · 11 | 1$^3$·4$^2$ | 2$^2$·4$^2$ |
| 5 = 5 | 1584 = 2$^4$ · 3$^2$ · 11 | 1·5$^2$ | 1$^2$·5$^2$ |
| 6 = 2 · 3 | 1320 = 2$^3$ · 3 · 5 · 11 | 2·3·6 | 1·2·3·6 |
| 8 = 2$^3$ | 990 = 2 · 3$^2$ · 5 · 11 | 1·2·8 | 4·8 | power equivalent |
| 8 = 2$^3$ | 990 = 2 · 3$^2$ · 5 · 11 | 1·2·8 | 4·8 | power equivalent |
| 11 = 11 | 720 = 2$^4$ · 3$^2$ · 5 | 11 | 1·11 | power equivalent |
| 11 = 11 | 720 = 2$^4$ · 3$^2$ · 5 | 11 | 1·11 | power equivalent |