# Semidihedral group

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## 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