Dicyclic Group Dic3/Matrix Representation

Matrix Representation of Dicyclic Group $\Dic 3$
Let $\omega$ denote the complex number $\map \exp {\dfrac {2 \pi i} 6}$, so that $\omega^6 = 1$.

Let $\mathbf X$ be the matrix defined as:
 * $\mathbf X = \begin{bmatrix} \omega & 0 \\ 0 & \omega^{-1} \end{bmatrix}$

Let $\mathbf Y$ be the matrix defined such that:
 * $\mathbf X \mathbf Y = \mathbf Y \mathbf X^{-1}$

and:
 * $\mathbf Y^2 = \mathbf X^3$

Then the set:
 * $G = \set {\mathbf X^i, \mathbf Y \mathbf X^j: 1 \le i, j \le 6}$

defines the dicyclic group $\Dic 3$.

Proof
By calculation, we have:

and it is seen that:


 * $\mathbf X^{-1} = \begin{bmatrix} \omega^{-1} & 0 \\ 0 & \omega \end{bmatrix}$

Now consider $\mathbf Y = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}$

We have:

and:

Thus $\mathbf X$ and $\mathbf Y$ fulfil the criteria given.