Semidihedral group

Definition

Let $n$ be a natural number greater than or equal to $4$. The semidihedral group or quasidihedral group of order $2^{n}$ (and degree $2^{n-1}$), denoted $SD_{2^{n}}$, is defined by the following presentation:

$\!SD_{2^{n}}=QD_{2^{n}}:=\langle a,x\mid a^{2^{n-1}}=x^{2}=e,xax=a^{2^{n-2}-1}\rangle$

(here, $e$ is the symbol for the identity element).

Particular cases

$n$	$2^{n}$	$n-1$	$2^{n-1}$	Group
4	16	3	8	semidihedral group:SD16
5	32	4	16	semidihedral group:SD32
6	64	5	32	semidihedral group:SD64
7	128	6	64	semidihedral group:SD128