Definition:Dicyclic Group

Definition
For even $n$, the dicyclic group $Q_n$ of order $2n$ is the group having the presentation:


 * $Q_n = \langle x,y \mid x^n = 1, y^2 = x^{\frac n 2}, y^{-1}xy = x^{-1}\rangle$

Quaternion Group
The group $Q_4$ is best known as the quaternion group.

The elements of $Q_4$ are $Q = \left\{{e, a, a^2, a^3, b, a b, a^2 b, a^3 b}\right\}$.

Its Cayley table is given by:

$\begin{array}{c|cccccccc} & e    & a     & a^2   & a^3   & b     & a b   & a^2 b & a^3 b \\ \hline e    & e     & a     & a^2   & a^3   & b     & a b   & a^2 b & a^3 b \\ a    & a     & a^2   & a^3   & e     & a b   & a^2 b & a^3 b & b     \\ a^2  & a^2   & a^3   & e     & a     & a^2 b & a^3 b & b     & a b   \\ a^3  & a^3   & e     & a     & a^2   & a^3 b & b     & a b   & a^2 b \\ b    & b     & a^3 b & a^2 b & a b   & a^2   & a     & e     & a^3   \\ a b  & a b   & b     & a^3 b & a^2 b & a^3   & a^2   & a     & e     \\ a^2 b & a^2 b & a b  & b     & a^3 b & e     & a^3   & a^2   & a     \\ a^3 b & a^3 b & a^2 b & a b  & b     & a     & e     & a^3   & a^2 \end{array}$

The Quaternion Group is Hamiltonian.