<s>[INST] Galois group of $x^6-2x^4+2x^2-2$ over $\mathbb{Q}$ 
Find Galois group of $x^6-2x^4+2x^2-2$ over $\mathbb{Q}$ and describe an extension corresponding to any of it's proper subgroups of maximal order.

I know that the roots are $$\sqrt{\frac{1}{3}\left(2 - \frac{2}{\sqrt[3]{17 + 3\sqrt{33}}} + \sqrt[3]{17 + 3 \sqrt{33}}\right)}$$ and $$\sqrt{\frac{2}{3}+\frac{1\mp3i}{3\sqrt[3]{17 + 3\sqrt{33}}}-\frac{1}{6}(1\pm i\sqrt3)\sqrt[3]{17 + 3\sqrt{33}}}$$
This looks a bit too complex to me. Can you please help me?

UPD: I know that Galois group of $x^3-2x^2+2x-2$ is $S_3$.

 [/INST] Hint: Write $x^6-2x^4+2x^2-2=g(x^2)$ and start by considering the spitting field of $g$ and its Galois group.
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