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
minimum size of generating set	2	groups with same order and minimum size of generating set | groups with same minimum size of generating set
number of conjugacy classes	10	groups with same order and number of conjugacy classes | groups with same number of conjugacy classes

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

Generating set as permutation representation

$M_{11}$ is isomorphic to the group generated by the permutations ${\sigma }_1$ and ${\sigma }_2$ where:

• ${\sigma }_1=(1,2,3,4,5,6,7,8,9,10,11)$

• ${\sigma }_2=(3,7,11,8)(4,10,5,6)$

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