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?



Representations of semidirect products of symmetric groups
The representations of wreath products of symmetric groups are known: for example, see section 4.3 of "The representation theory of the symmetric group" by James and Kerber. 


