Definition:Dicyclic Group

Definition
For even $n$, the dicyclic group $\Dic n$ of order $4 n$ is the group having the presentation:


 * $\Dic n = \gen {x, y: x^{2 n} = e, y^2 = x^n, y^{-1} x y = x^{-1} }$

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


 * $Q_{2 n} = \gen {x, y: x^{2n} = e, y^2 = x^n, y^{-1} x y = x^{-1} }$

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


 * $Q_4 = \gen {x, y: x^4 = e, y^2 = x^2, y^{-1} x y = x^{-1} }$

Others have a different notation again:


 * $Q_{4 n} = \gen {x, y: x^{2 n} = e, y^2 = x^n, y^{-1} x y = x^{-1} }$

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


 * $Q_8 = \gen {x, y: x^4 = e, y^2 = x^2, y^{-1} x y = x^{-1} }$

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 denotes $Q$) $\Dic n$ is used throughout.