If $G$ is a group and $H$ is a subgroup of $S_n$ we can form their wreath product $G \wr H = \{(g_1, ..., g_n; \pi): g_i \in G$ and $\pi \in H\}$. I'm wondering whether the following is correct:

 1. $<(e, ..., g_i, ..., e; e)> = \{(g_1, ..., g_n; e)\}$

 2. $\{(g_1, ..., g_n; e)\} \circ \{(e, ..., e; \pi)\} = G \wr H$

Additionally, will all elements of $G \wr H$ be generated distinctly? Or in other words is it possible to do this more efficiently?