Definition:Dicyclic Group

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


 * $Dic_n = \left \langle{x, y: x^{2 n} = e, y^2 = x^n, y^{-1} x y = x^{-1}}\right \rangle$

Alternative Notation
Some sources denote the group $Dic_n$ as $Q_{2n}$, referring to it as the generalized quaternion group:


 * $Q_{2n} = \left \langle{x, y: x^{2n} = e, y^2 = x^n, y^{-1}xy = x^{-1} }\right \rangle$

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


 * $Q_4 = \left \langle{x, y: x^4 = e, y^2 = x^2, y^{-1}xy = x^{-1} }\right \rangle$

Others have a different notation again:


 * $Q_{4n} = \left \langle{x, y: x^{2n} = e, y^2 = x^n, y^{-1}xy = x^{-1} }\right \rangle$

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


 * $Q_8 = \left \langle{x, y: x^4 = e, y^2 = x^2, y^{-1}xy = x^{-1} }\right \rangle$

Because of the potential ambiguity, it is recommended that $Q_{2n}$ and $Q_{4n}$ are not used, but that $Dic_n$ is used throughout.