# Mathieu group:M24

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## Definition

This is the Mathieu group of degree 24, denoted $M_{24}$, and is the subgroup of the symmetric group of degree 24 generated by the following permutations:

$M_{24} := \langle (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23), (3,17,10,7,9)(4,13,14,19,5)(8,18,11,12,23)(15,20,22,21,16), (1,24)(2,23)(3,12)(4,16)(5,18)(6,10)(7,20)(8,14)(9,21)(11,17)(13,22)(15,19) \rangle$

## Arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 244823040 groups with same order
exponent of a group 212520 groups with same order and exponent of a group | groups with same exponent of a group

### Arithmetic functions of a counting nature

Function Value Explanation
number of conjugacy classes 26

## Group properties

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

## GAP implementation

GAP's SmallGroup library is not available for this large order.

Description Functions used
MathieuGroup(24) MathieuGroup