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:

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

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

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