How to find linear fractional transformations of a group

Question

Finite presentation of G is $$\langle x,y,t,q : x^2=y^3=t^2=q^2 =1,tq=qt,ty=yt,qyq=y^{−1},xt =qx \rangle.$$ I am interested in finding linear fractional transformations x,y,t,q which satisfy all its relations.

Answer 1

Let $H<G$ be generated by $y, t, q$. It is isomorphic to the dihedral group $G_2$ of order $12$; the latter embeds in $PU(2)< PSL(2,C)$ Furthermore, the (unique up to conjugation) monomorphism
$$ r: H \to G_2< PU(2) $$ extends to the rest of $G$, since order 2 elements $r(t)$ and $r(s)$ are conjugate (in $PU(2)$). I do not know if this is what you are looking for. (Maybe you want $r$ to be also injective, this would require more work and I do not see much motivation.)