$E$ is the splitting field for $f(x)=x^3-2$ and $K := E(\sqrt 5)$. We want to show that $G=\text{Gal}(K/\mathbb{Q}) \cong \mathbb{Z_2} \times S_3$.
To do this, we know that $K$ has subfields $E$ and $F=\mathbb{Q}(\sqrt 5)$, which are Galois extensions of $\mathbb{Q}$. Hence, there are restriction maps given as $$r_E: G\rightarrow \text{Gal}(E/\mathbb{Q}) ~~~ \text{and} ~~~ r_F: G\rightarrow \text{Gal}(F/\mathbb{Q})$$
We know $\text{Gal}(E/\mathbb{Q}) \cong S_3$ and $\text{Gal}(F/\mathbb{Q}) \cong \mathbb{Z_2}$ so it suffices to show that the map $G \rightarrow \text{Gal}(E/\mathbb{Q}) \times \text{Gal}(F/\mathbb{Q})$ given by $\sigma \mapsto (r_E(\sigma),r_F(\sigma))$ is an isomorphism.
I am not sure how I would go about showing that this map defines an isomorphism. Any suggestions?