Outer linear group

Definition

The outer linear group of degree $n$ over a commutative unital ring $R$ is defined as the external semidirect product of the general linear group $G L (n, R)$ with a cyclic group of order two, where the non-identity element of the cyclic group acts by the transpose-inverse map.

Size of field	Degree (order of matrices)	Common name for the outer linear group	Order of group	Comment
$q$	1	Dihedral group $D_{2 (q - 1)}$	$2 (q - 1)$	Multiplicative group of field is cyclic of order $q - 1$ , outer automorphism acts by inverse map.
$2^{n}$	2	Direct product of $S L (2, 2^{n})$ and dihedral group $D_{2 (2^{n} - 1)}$
2	1	Cyclic group:Z2	$2$
2	2	Dihedral group:D12 (also, direct product of $S_{3}$ and $Z_{2}$	$12 = 2^{2} \cdot 3$	supersolvable but not nilpotent.
3	2	Outer linear group:OL(2,3)	$96 = 2^{5} \cdot 3$	solvable
4	2	Direct product of A5 and S3	$360 = 2^{3} \cdot 3^{2} \cdot 5$
5	2	Outer linear group:OL(2,5)	$960 = 2^{6} \cdot 3 \cdot 5$
2	3	Projective general linear group:PGL(2,7)	$336 = 2^{4} \cdot 3 \cdot 7$