$G$ 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 $S_3$ ??

find an element $a$ of order 2 ,an element $b$ of order 3 and check whether $ab=b^2a$ holds
