Using wreath products to find stabilisers of a partition of a set I have the following example, that uses the wreath product to find the stabilisers of a partition. I don't understand how the wreath product does this though. I can recite the definition of a wreath product, but I think this comes down to me not having any intuition about what it does.
Given a partition $\scr{P} = \{\{1,2,3\},\{4,5,6\},\{7,8\}\}$ of 8 elements, we can find the elements of $\mathrm{Sym}(8)$ that stabilise it via $$\mathrm{Stab}_G(\mathscr{P}) = \mathrm{Sym}(\{1,2,3\})\wr \mathrm{Sym}(2) \times \mathrm{Sym}(\{7,8\}).$$
Could someone please explain to me how this makes sense? I can see some of the groups there are the stabilisers of some parts of $\mathscr{P}$, but I fail to see how the wreath product with $\mathrm{Sym}(2)$ produces all stabilisers?
 A: The difficulty I have with your question is that for me the stabilizer of a partition of a set of size $mn$ into $n$ sets of size $m$ is actually the definition of the permutation wreath product $S_m \wr S_n$
Following from the definition, you can then derive its structure as a semidirect product $S_m^n \rtimes S_n$, which is essentially what you have done yourself already.
There is a normal subgroup (the base group) isomorphic to $S_m^n$ consisting of all permutations that stabilize each of the sets in the partitition. This has a complement isomorphic to $S_n$, which permutes the $n$ sets. (In fact it has many such complements.)
So to give another example, the stabilizer of the partition $\{\{1,2,3\},\{4,5,6\},\{7,8,9\}\}$ is $S_3 \wr S_3 = S_3^3 \rtimes S_3$.
The base group is generated by $(1,2,3),(2,3),(4,5,6),(5,6),(7,8,9),(8,9)$. One complement of this is generated by $(1,4,7)(2,5,8)(3,6,9),\,(4,7)(5,8)(6,9)$, but there are many other complements, such as $\langle(1,5,9)(2,6,7)(3,4,8),\,(5,9)(6,7)(4,8)\rangle$. 
Since it is a semidirect product, a general element of the wreath product is a product from an element of the base group with an element of one specific complement.
To see this, suppose we are given some random element that stabilizes the partition, such as $g=(1, 9, 5, 2, 7, 6, 3, 8, 4)$. This induces the permutation $(1,3,2)$ on the three sets in the partition, and the element in our first complement that induces that permutation is $h=(1,7,4)(2,8,5)(3,9,6)$. So we should have $g = kh$ for some $k$ in the base group, and you can calculate $k =(4, 6, 5)(7, 9, 8)$ (composing permutations from right to left).
This is probably not a completely satisfactory answer to your question, but I hope it will clarify thing sto some extent.
