Splitting field is $K=\mathbb Q (\sqrt[4]{-5}, i)$
Degree of $K$ over rationals is $8$ so the galois group $G=\text{G}(K/ \mathbb Q)$ has order $8$.
$x^4 +5$ is irreducible so there is one orbit which is the roots set $$R=\{ \sqrt[4]5 \xi, \sqrt[4]5 \xi^3, \sqrt[4]5 \xi^5, \sqrt[4]5 \xi^7 \}$$ where $\xi = e^{\frac{\pi}4 i}$.
Faithful action implies that $G \leq S_4$ and $D_8$ is the only subgroup of $S_4$ which has order $8$ so this is the isomorphism type of $G$.
But there are three copies of $D_8$, which one do we use to determine the action on the generators of $K$?