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 $S D_{2^{n}}$ , is defined by the following presentation:

$S D_{2^{n}} = Q D_{2^{n}} : = ⟨ a, x ∣ a^{2^{n - 1}} = x^{2} = e, x a x = a^{2^{n - 2} - 1} ⟩$

(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