I know if I have a polynomial $f(x) = g(x) \cdot h(x) \cdot k(x)$ and $g(x),h(x),k(x)$ are polynomials of degree 2, then the Galois group will be $Z/2Z \times Z/2Z \times Z/2Z$ if the roots of $g,h,k$ are distinct.
But what happens in the case when you have mixed factors... i.e. for example the polynomial
$(x^2 + 1)(x^2 -4x + 7)(x^2 - 2)$
the splitting field would be $Q(\sqrt{2},\sqrt{3},i)$ but what would be the Galois group?