$G$ is isomorphic to $S_3$ Show that the Galois group of the splitting field $F$ of $X^3-7$ over $\mathbb{Q}$ is isomorphic to $S_3$. 
I have found that the the Galois group is the following: 
$$G=\{\tau_{ij}, i=1,2,3, j=1,2\}$$ 
where $$\tau_{ij}(\sqrt[3]{7})=\omega^{i-1}\sqrt[3]{7} \\ \tau_{ij}(\omega)=\omega^j \\ i=1,2,3 , j=1,2$$ 
Is this correct?? 
How could I continue to show that $G$ is isomorphic to $S_3$ ??

First observe that $|G|=6$, because you've written down 6 automorphisms and we know $|G|\leq 3!$. Next observe $G$ is not cyclic, because each of the $\tau_{ij}$ have order $<6$. Thus $G=S_3$.
