I would like to find Galois group of $P(X)=X^6−4X^3+2$.

I have that $(2+\sqrt{2})^{1/3}$ is a root of $P$, denote it $\alpha$. By Eisenstein $P(X)$ is irreducible over $\Bbb{Q}$ then $[\Bbb{Q}(\alpha):\Bbb{Q}]=6.$ 

If we let $\beta:=(2-\sqrt{2})^{1/3}$ then the root of $P(X)$ are $\alpha,\omega\alpha,\omega^2\alpha,\beta,\omega\beta,\omega^2\beta$ where $\omega$ is non trivial $3-$root of unity; then $\Bbb{Q(\alpha,\beta,\omega)}$ is the splitting field of $P(X)$, noted $E.$ It seems that $[E:\Bbb{Q}]=36.$

Not sure how can I find the Galois group ?