Definition:Dicyclic Group

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

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

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 group product 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.