Groups of Order 21/Matrix Representation of Non-Abelian Instance

Theorem
Let $G$ be the group of order $21$ whose group presentation is:
 * $\gen {x, y: x^7 = e = y^3, y x y^{-1} = x^2}$

Then $G$ can be instantiated by the following pair of matrices over $\Z_7$:


 * $X = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \qquad Y = \begin{pmatrix} 4 & 0 \\ 0 & 2 \end{pmatrix}$

Proof
We calculate the powers of $X$ and $Y$ in turn:

and so on to:

Thus we have:
 * $X^7 = \mathbf I$

where $\mathbf I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ is the identity matrix.

Then:

Thus we have:
 * $Y^3 = \mathbf I$

and:
 * $Y^{-1} = \begin{pmatrix} 2 & 0 \\ 0 & 4 \end{pmatrix}$.

Then:

and the result is apparent.