I need to show that the subgroup of $GL(2,\mathbb{C})$ generated by $$a:=\begin{pmatrix}w^2 & 0 \\ 0 & w\end{pmatrix}$$ where $w=e^{\frac{2\pi i}{3}}$ and $$b:=\begin{pmatrix}0 & i \\i & 0\end{pmatrix}$$ is isomorphic to the presentation $$\langle x,y\mid x^3,y^4,yx=x^2y\rangle .$$
I don't even know where to start. Any help would be appreciated. Thanks.
Let $F=\langle x,y \rangle$ be the free group of two generator,define a homomorphism $\phi$ be declaring $\phi(x)=a$ and $\phi(y)=b$ extending as a homomorphism. We just have to show that the kernel is the group generated by the relations. It is clear that $\langle x^3,y^4, x^2yx^{-1}y^{-1} \rangle $ is in the kernel.
To show that other inclusion observe that if an expression $x^{s_1}y^{s_2} \ldots y^{s_n}$. is in the kernel then using the relation $yx=x^2y$ one can bring the relation to the form $x^my^n$. Then show that in this case 3 divides $m$ and $4$ divides $n$ completing the proof.