Let $\mathcal{P} = \{ A_1, \ldots, A_k \}$ be a set partition of $[nk] := \{ 1,\ldots,nk \}$ into $k$ blocks of $n$ elements. This means that:
- For all $i$, $A_i \subset [nk]$ and $|A_i| = n$;
- if $i \neq j$, $A_i \cap A_j = \emptyset$;
- $\bigcup_{i=1}^k A_i = [nk]$.
Example: Take $n=4$, $k=3$, and $\mathcal{P} = \{ \{1,2,3,4\}, \{5,6,7,8\}, \{9,10,11,12\} \}$.
I want to know how to write the subgroup of the symmetric group $S_{[n]}$ which leaves $\mathcal{P}$ invariant. Of course, this subgroup contains $S_{A_1} \times \ldots \times S_{A_k}$, but it also contains permutations that completely exchange two blocks.
In the example, such an "other" permutation is $(1,5)(2,6)(3,7)(4,8)$, which exchanges the first two blocks.
At first, I thought that I could write it as the semidirect product $\left( S_{A_1} \times \ldots \times S_{A_k} \right) \rtimes S_k$, where $S_k$ acts on the indices. But as this action is not an automorphism of $S_{A_1} \times \ldots \times S_{A_k}$ (permuting the factors gives an isomorphic group, but not the same group), it does not correspond to the definition of a semidirect product. What is the good definition for this construction?