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.

Group properties

Property	Satisfied	Explanation
Abelian group	No
Nilpotent group	Yes
Solvable group	Yes
Supersolvable group	Yes
Metacyclic group	Yes
Ambivalent group	Yes
Rational group	Yes only for $k=2$, i.e., the quaternion group

Examples

Small values

$k$	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