How can I determine whether $\prod^t_i S_{N_i} \wr D_{m_i}$ a normal subgroup of $S_{\sum^t_i N_i m_i}$?
Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ symbols, $D_{m_i}$ is the dihedral group of order $2 m_i$.
My effort:
I understand that there are more than one ways to determine this without checking all group elements.
Construct a homomorphism having it as kernel
Here we have to construct a homomorphism $\varphi: S_{\sum^t_i N_i m_i} \to K$ such that the kernel of the homomorphism is precisely $\prod^t_i S_{N_i} \wr D_{m_i}$. How can I do that?
Compute its commutator with the whole group
How can I check if $[S_{\sum^t_i N_i m_i}, \prod^t_i S_{N_i} \wr D_{m_i}]$ is contained in $\prod^t_i S_{N_i} \wr D_{m_i}$ ?