# Projective special linear group

## Particular cases

### Finite fields

Some facts:

Size of field Order of matrices Common name for the projective special linear group Order of group Comment
$q$ 1 Trivial group $1$ Trivial
2 2 Symmetric group:S3 $6 = 2 \cdot 3$ supersolvable but not nilpotent. Not simple.
3 2 Alternating group:A4 $12 = 2^2 \cdot 3$ solvable but not supersolvable group. Not simple.
4 2 Alternating group:A5 $60 = 2^2 \cdot 3 \cdot 5$ simple non-abelian group of smallest order.
5 2 Alternating group:A5 $60 = 2^2 \cdot 3 \cdot 5$ simple non-abelian group of smallest order.
7 2 Projective special linear group:PSL(3,2) $168 = 2^3 \cdot 3 \cdot 7$ simple non-abelian group of second smallest order.
9 2 Alternating group:A6 $360 = 2^3 \cdot 3^2 \cdot 5$ simple non-abelian group.
2 3 Projective special linear group:PSL(3,2) $168 = 2^3 \cdot 3 \cdot 7$ simple non-abelian group of second smallest order.
3 3 Projective special linear group:PSL(3,3) $5616 = 2^4 \cdot 3^3 \cdot 13$ simple non-abelian group.