Projective special linear group:PSL(3,4)

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Definition

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

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