Splitting field and Galois group of $(x^5-1)(x^2+1)$ over $\mathbb{Q}$

Consider $p(x) = (x^5-1)(x^2+1)$. Then, its splitting field is $\mathbb{Q}(e^{\frac{2\pi i}{5}}, i)$.

Thus, $f\in \text{Gal}(\mathbb{Q}(e^{\frac{2\pi i}{5}}, i)/\mathbb{Q})$ maps $\omega = e^{\frac{2\pi i}{5}}$ to any of $\omega^k$ for $k=1,...,4$ and $i$ to $\pm i$.

In that way, I can conclude that $|\text{Gal}(\mathbb{Q}(e^{\frac{2\pi i}{5}}, i)/\mathbb{Q})|= 4 \cdot 2 = 8$.

Now, how do I know which of the groups of order $8$ it is? It might be $\mathbb{Z}_2 \times \mathbb{Z}_4$ because of element orders but I am not sure.

from Hot Weekly Questions - Mathematics Stack Exchange