Semi-dihedral group (awaiting review)

The semi-dihedral group $\SD_{2^k}$ is one of the four non-abelian groups of order $2^k$ containing a cyclic subgroup of index 2 (the other three being the dihedral group, the (generalized) quaternion group, and the other-dihedral group). It has the group presentation: $$\SD_{2^k} = \langle r,s\mid r^{2^{k-1}}=s^{2}=1,\ srs=r^{2^{k-2}-1} \rangle.$$ It can also be given as the semidirect product $C_{2^{k-1}} \rtimes C_2$ where the action of $C_2$ on $C_{2^{k-1}}$ is given by $x \mapsto x^{2^{k-2}-1}$.