# 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