Mathieu group:M11
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Contents
Definition
In terms of 
This group, termed the Mathieu group of degree eleven and denoted 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.
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 . There are also Mathieu groups for parameters
, but these are not simple groups. The Mathieu group for parameter
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 ![]() ![]() |
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: ![]() ![]() ![]() |
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 ![]() ![]() |
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 |