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:

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

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

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

The generalized quaternion group is generally only ever defined for $k\geq 2$. However, if we put $k=1$ we retrieve the Klein four-group.

Group properties

Property	Satisfied	Explanation
Abelian group	No
Nilpotent group	Yes. Nilpotency class $k$
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