Projective special linear group:PSL(3,4)

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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 group is a finite group defined in the following equivalent ways:

• As the projective special linear group of degree three over the field of four elements.
• As the Mathieu group of degree $21$.

It is a member of the smallest pair of distinct isomorphism classes of finite simple non-abelian groups that have the same order; the other member of the pair being alternating group:A8, which is also $PSL(4,2)$. See there are at most two finite simple groups of any order for more information.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 20160#Arithmetic functions

Basic arithmetic functions

Function	Value	Similar groups	Explanation
order (number of elements, equivalently, cardinality or size of underlying set)	20160	groups with same order	As $PSL(3,q),q=4$: $q^{3}(q^{3}-1)(q^{2}-1)/\operatorname {gcd} (3,q-1)=4^{3}(4^{3}-1)(4^{2}-1)/\operatorname {gcd} (3,3)=(64)(63)(15)/3=20160$

Arithmetic functions of a counting nature

Function	Value	Similar groups	Explanation
number of conjugacy classes	10	groups with same order and number of conjugacy classes | groups with same number of conjugacy classes	As $PSL(3,q),q=4$ (case $q$ is 1 mod 3): $(q^{2}+q+10)/3=(4^{2}+4+10)/3=10$ See element structure of projective special linear group of degree three over a finite field for more information

GAP implementation

Description	Functions used
PSL(3,4)	PSL
MathieuGroup(21)	MathieuGroup