TITLE: Generators of $GL(2, \mathbb{F}_3)$
QUESTION [5 upvotes]: I came across the statement that for $GL(2, \mathbb{F}_3)$ we can choose generators $x$, $y$, $z$, that satisfy relations $x^4=y^3=z^2=xyz$, $(xyz)^2=1$. What matrices can I take for $x$, $y$, $z$?

REPLY [2 votes]: I just remark that ${\rm GL}(2,3)$ has a presentation $\langle x,y,z : x^8 = y^3 = z^2 = (xyz) = [x^4,y ] = 1 \rangle.$ This presentation gives a double cover of $S_4,$ but one which contains more than one involution, since $z \neq x^4.$
You can take $z = \left( \begin{array}{crclcr} 1 & 0\\ 0 & 2 \end{array} \right)$ and $y = \left( \begin{array}{clcr} 2 & 2\\1 & 0 \end{array} \right) .$