Definition:Dicyclic Group/Quaternion Group

Definition
The dicyclic group $Q_4$ is known as the quaternion group.

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

Its group presentation is given by:
 * $Q_4 = \left \langle {a, b: a^4 = e, b^2 = a^2, a b a = b}\right \rangle$

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}$

Some sources refer to this group merely as $Q$.

Also see
The Quaternion Group is Hamiltonian.