The quaternion group

A typical exercise in elementary group theory asks to check that the group generated by certain two $2\times 2$ complex matrices is isomorphic to the quaternion group.

Let us check that the group generated by $\begin{pmatrix} \sqrt{-1}&0 \\ 0&-\sqrt{-1} \end{pmatrix}$ and $\begin{pmatrix} 0&1 \\ -1&0 \end{pmatrix}$ is isomorphic to the quaternion group $Q_8$. To construct the group we proceed as follows:

> K<i> := CyclotomicField(4);
> a := Matrix(2, [i,0,0,-i]);
> b := Matrix(2, [0,1,-1,0]);
> G := MatrixGroup<2,K|a,b>;
> GroupName(G);
Q8