I am trying to interpret $S_{N} \wr D_{m}$ in the light of the interpretation of $\mathbb{Z}^n_2 \wr \mathbb{Z}_2$ in this paper.
So, according to the definition,
$$\mathbb{Z}^n_2 \wr \mathbb{Z}_2 = (\mathbb{Z}^n_2 \times \mathbb{Z}^n_2) \rtimes \mathbb{Z}_2 $$
Here, $N = (\mathbb{Z}^n_2 \times \mathbb{Z}^n_2)$ is the group of ordered pairs of binary strings of length $n$. And $N \rtimes \mathbb{Z}_2$ is the group of ordered pairs of elements from $(\mathbb{Z}^n_2 \times \mathbb{Z}^n_2)$ with $0$ or $1$.
Assuming my interpretation is correct here is my attempt to interpret $S_{N} \wr D_{m}$.
$S_{N} \wr D_{m}$ is the group of ordered pairs of elements from $S_N$ with the elements of $D_m$.
Am I doing it right?