Definition:Dicyclic Group/Also denoted as

Dicyclic Group: Also denoted as

Some sources denote the dicyclic group $\Dic n$ as $Q_{2 n}$, referring to it as the generalized quaternion group:

$Q_{2 n} = \gen {a, b: a^{2 n} = e, b^2 = a^n, b^{-1} a b = a^{-1} }$

Using this notation, it can be seen that the quaternion group is represented by:

$Q_4 = \gen {a, b: a^4 = e, b^2 = a^2, b^{-1} a b = a^{-1} }$

Others have a different notation again:

$Q_{4 n} = \gen {a, b: a^{2 n} = e, b^2 = a^n, b^{-1} a b = a^{-1} }$

Using this notation, it can be seen that the quaternion group is represented by:

$Q_8 = \gen {a, b: a^4 = e, b^2 = a^2, b^{-1} a b = a^{-1} }$

Because of the potential ambiguity, it is recommended that $Q_{2 n}$ and $Q_{4 n}$ are not used, but that (except for the quaternion group itself, which $\mathsf{Pr} \infty \mathsf{fWiki}$ denotes $Q$) $\Dic n$ is used throughout.