Semidihedral group

Definition

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

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

(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