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).