Let a group be $\prod^t_{i=1} S_{N_i} \wr D_{m_i} $ where $t \in \mathbb{Z} \text{ and } t \ge 1; N_i, m_i \in \mathbb{Z} \text{ and } N_i, m_i \ge 1$; and $S_{N_i}$ is a symmetric group over $N_i$ symbols and $D_{m_i}$ is the dhedral group of order $2 m_i$.
My question: How do I compute the characters of $\prod^t_{i=1} S_{N_i} \wr D_{m_i} $?
An example: Let $t = 2, N_1 = 3, N_2 = 4, m_1 = 5, \text{ and } m_2 = 6$. So, how do I compute the characters of the following group?
$$ \prod^2_{i=1} S_{N_i} \wr D_{m_i} \\ = \left(S_{N_1} \wr D_{m_1}\right) \times \left(S_{N_2} \wr D_{m_2}\right)\\ = \left(S_3 \wr D_5\right) \times \left(S_4 \wr D_6\right)\\ = \left(\left( S_3 \right)^5 \rtimes D_5\right) \times \left(\left( S_4 \right)^6 \rtimes D_6\right)\\ = \left(\left( S_3 \times S_3 \times S_3 \times S_3 \times S_3 \right) \rtimes D_5\right) \times \left(\left( S_4 \times S_4 \times S_4 \times S_4 \times S_4 \times S_4 \right) \rtimes D_6\right) $$
So, how do I compute the characters of $\left(\left( S_3 \times S_3 \times S_3 \times S_3 \times S_3 \right) \rtimes D_5\right) \times \left(\left( S_4 \times S_4 \times S_4 \times S_4 \times S_4 \times S_4 \right) \rtimes D_6\right)$?