Mathieu group:M22

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

This is the Mathieu group of degree 22, denoted $M_{22}$, and is the subgroup of the symmetric group of degree 22 generated by the following permutations:

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

Arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	443520	groups with same order
exponent of a group	9240	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	12

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(22)	MathieuGroup