How can I determine the order of $\sum^t_i S_{N_i} \wr D_{m_i}$ ?
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 the order of semidirect product is product of orders of the groups. So, $|S_{N_i} \wr D_{m_i}| = |S_{N_i}| |D_{m_i}|$. For non-Abelian group direct sum and direct product are the same. So, in that sense the order is determined as follows. $$ |\sum^t_i S_{N_i} \wr D_{m_i}| = \Pi_i |S_{N_i} \wr D_{m_i}| $$
Am I doing it right?