The Galois group of a specific polynomial $ f(x) = x^6-2x^3+2 \in Q[x] $ Hello all I was given this question in Algebra asking me to show a polynomial's Galois group has both a subgroup and quotient group isomorphic to $S_3$ the three symmetric group.
The polynomial in question over Q is $ f(x) = x^6-2x^3+2 \in Q[x] $
I know this polynomial is irreducible over Q via Eisenstein with p=2
This I know so the polynomial is irreducible over a field Q with char zero therefore it is separable, so it has a Galois group for its splitting field.
I also know this is the minimal polynomial of $ \sqrt[3]{1+i} $ over Q. I cannot seem to proceed and actually prove that the Galois group has both a subgroup and quotient groups isomorphic to $ S_3 $, I also know the Galois group is a subgroup of the six symmetric group $S_6$ but cannot use any of this information to prove the claim. Help appreciated thanks all

Hint: What is the dimension of the extension field of f(x) over $\mathbb{Q}$?
