I have shown that if $x$ is the nontrivial cubic root of the unity and that if $y$ is the real cubic root of $2$, then $Q(x,y)$ is a Galois extension whose Galois group has order $6$.
I know that the Galois group is determined by the automorfisms:
$α↦ωα,ω↦ω,α↦ωα,ω↦ω^{-1},α↦ω^2α,ω↦ω$,
$α↦ω^2α,ω↦ω ^{-1},α↦ω^3α,ω↦ω,α↦ω^3α,ω↦ω^{-1}$
The question is how can I make explicit the isomorphism between this Galois group and $S_3$? drawing the Galois correspondence?