Splitting field and intermediate fields

Question

Find splitting field $K$ of polynomial $x^3-7$ over $\mathbb{Q} (\sqrt{3} )$ and find

1. $(K: \mathbb{Q} )$
2. $G(K/ \mathbb{Q} )$
3. Find intermediate fields between $\mathbb{Q}$ and $K$.

So the splitting field is $K=\mathbb{Q} ( \sqrt{3}, \epsilon _3, \sqrt[3]{7}) = \mathbb{Q} (\sqrt{3},i,\sqrt[3]{7})$ and

1. $(K: \mathbb{Q} ) = 2 \cdot 2 \cdot 3 =12$
2. $G(K/ \mathbb{Q} ) = \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3$ as $\sqrt{3} \mapsto \pm \sqrt{3} $ , $i \mapsto \pm i$, $\sqrt[3]{7} \mapsto \sqrt[3]{7} $ or $\sqrt[3]{7} \mapsto \epsilon _3 \sqrt[3]{7}$ or $\sqrt[3]{7} \mapsto \epsilon _3 ^2 \sqrt[3]{7}$.

Can someone check if I am right? Thanks in advance :)