Generalized quaternion group

Definition

A generalized quaternion group is a group of order $2^{k+1}$ with generators $x$ and $a$ such that the group has the presentation:

$<a,x|x^{2}=a^{2^{k}}=1,xax^{-1}=a^{-1}>$

Equivalently, it is the dicyclic group with parameter $2^{k-1}$.

For the particular case $k=2$, we recover the quaternion group.