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:

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

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

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

$k$	Group	Order, $2^{k + 1}$
2	quaternion group	8
3	generalized quaternion group:Q16	16
4	generalized quaternion group:Q32	32
5	generalized quaternion group:Q64	64
6	generalized quaternion group:Q128	128
7	generalized quaternion group:Q256	256
8	generalized quaternion group:Q512	512
9	generalized quaternion group:Q1024	1024
10	generalized quaternion group:Q2048	2048