This is sort of a vague (I apologize in advance) question, but I'm interested in the representation theory of the following group

$A \rtimes B$, where $A = (S_1)^{m_1} \times (S_2)^{m_2} \times \ldots \times (S_r)^{m_r}$, $B = S_{m_1} \times S_{m_2} \times ... \times S_{m_r}$, and $B$ acts on $A$ by permuting the factors. Is something nice known about the representation theory of these groups? Does anyone know a good reference for something like this?

wreath productsof $S_i$ with $S_{m_i}$. Check out en.wikipedia.org/wiki/Wreath_product; I'm sure a lot is known about the representation theory of wreath products of symmetric groups. $\endgroup$3more comments