# Mathieu group:M11

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## Definition

### In terms of $M_{12}$

This group, termed the Mathieu group of degree eleven and denoted $M_{11}$ is the subgroup of the symmetric group of degree eleven defined as the isotropy subgroup of any point under the natural action of Mathieu group:M12 on the projective line over field:F11.

$M_{11}$ is in fact a subgroup of the alternating group of degree eleven.

### Relation with Mathieu groups

This is one of the five simple Mathieu groups, which form a subset of the sporadic simple groups. The parameters for the simple Mathieu groups are $11, 12, 22, 23, 24$. There are also Mathieu groups for parameters $9,10$, but these are not simple groups. The Mathieu group for parameter $21$ is a simple group that is not a sporadic simple group; it is isomorphic to the projective special linear group:PSL(3,4).

## Arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 7920 groups with same order As Mathieu group $M_n, n = 11, n \in \{ 9,10,11,12 \}$: $n!/7! = n(n - 1) \dots 8 = (11)(10)(9)(8) = 7920$
exponent of a group 1320 groups with same order and exponent of a group | groups with same exponent of a group
Frattini length 1 groups with same order and Frattini length | groups with same Frattini length

## Group properties

Property Satisfied? Explanation
abelian group No
nilpotent group No
solvable group No
simple group Yes
minimal simple group No

## Subgroups

Further information: subgroup structure of Mathieu group:M11

### Quick summary

Item Value
number of subgroups 8651
number of conjugacy classes of subgroups 39
isomorphism classes of Sylow subgroups and corresponding Sylow numbers 2-Sylow: semidihedral group:SD16, Sylow number is 495
3-Sylow: elementary abelian group:E9, Sylow number is 55
5-Sylow: cyclic group:Z5, Sylow number is 396
11-Sylow: cyclic group:Z11, Sylow number is 144
Hall subgroups apart from the whole group, trivial subgroup, and Sylow subgroups, there exist the following Hall subgroups: $\{ 2,3 \}$-Hall (order 144), $\{ 5, 11\}$-Hall (order 55), $\{2,3,5 \}$-Hall (order 720)
maximal subgroups maximal subgroups of order 48, 120, 144, 660, 720
normal subgroups the group is simple non-abelian, so the only normal subgroups are the whole group and the trivial subgroup
subgroups that are simple non-abelian groups (apart from the whole group) alternating group:A5 (order 60), alternating group:A6 (order 360), projective special linear group:PSL(2,11) (order 660)

## Linear representation theory

Further information: linear representation theory of Mathieu group:M11

### Summary

Item Value
degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$) 1,10,10,10,11,16,16,44,45,55
grouped form: 1 (1 time), 10 (3 times), 11 (1 time), 16 (2 times), 44 (1 time), 45 (1 time), 55 (1 time)
maximum: 55, quasirandom degree: 10, number: 10, lcm: 7920, sum of squares: 7920

## GAP implementation

The Mathieu group has order 7920. Unfortunately, GAP's SmallGroup library is not available for this order. The group can be constructed in either of these ways:

Description Functions used
MathieuGroup(11) MathieuGroup
PerfectGroup(7920) or equivalently PerfectGroup(7920,1) PerfectGroup