Consider the subgroup $G$ of $GL_{2}(\mathbb{C})$ generated by $A=\begin{pmatrix} \omega & 0 \\ 0 & \omega^{2} \end{pmatrix}$ and $B=\begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$ where $\omega=e^{\frac{2\pi i}{3}}$. Is there an isomorphism between $G$ and $H:=\langle a\in A,b\in B|a^{6}=I,b^{2}=a^{3}=(ab)^{2}\rangle$?
I computed that $A^3=B^4=I$ so is this enough so prove that $G$ is of order $12$? And I have very little intuition how to show the isomorphism. Probably by computing its 2-Sylow subgroup and whether it is a normal subgroup?