# Mathieu group:M11

View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Definition

This group, termed the Mathieu group of degree eleven and denoted $M_{11}$ is the subgroup of the symmetric group of degree eleven defined by the following generating set:

$\! M_{11} = \langle (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) \rangle$.

Note that since both the generating permutations are even permutations, $M_{11}$ is in fact a subgroup of the alternating group of degree eleven.

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 sporadic 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
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

## GAP implementation

### Definition using the Mathieu group function

The Mathieu group has order $7920$. Unfortunately, GAP does not assign group IDs for groups of such large orders. However, this group can be defined using the MathieuGroup function, as:

MathieuGroup(11)