Two element presentation of $\mathrm{GL}(2,3).$

Question

I am tying to find two element presentation of $\mathrm{GL}(2,3)$ with $48$ elements. I am trying to prove (which may be wrong ) that presentation of $\mathrm{GL}(2,3)$ is $\{a,b\mid a^3,b^2,(ab)^8\}$ where $a=\begin{bmatrix} 2 & 1 \\ 2 & 0 \end{bmatrix}$ and $b=\begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}.$ I also tried with GAP as

gap> f:=FreeGroup("a","b");;
gap> g:=f/[f.1^2,f.2^3,(f.1*f.2)^8];
<fp group on the generators [ a, b ]>
gap>Size(g);
#I  Coset table calculation failed -- trying with bigger table limit

Its does not calculate the order either. I Don't know GAP properly. Please help me to find its presentation. Thanks in advance.

Answer 1

Your current presentation is for an infinite group, which is why GAP runs out of space.

Your guess was close though. The following is a correct presentation.

$\langle a,b\mid a^2,b^3,(ab)^8,(ab)^4(ab^{-1})^4 \rangle$

You can verify this with the following GAP commands:

f:=FreeGroup("a","b");
a:=f.1;
b:=f.2;
g:=f/[a^2,b^3,(a*b)^8,(a*b)^4*(a*b^-1)^4];
Size(g);
StructureDescription(g);