TITLE: Centralizer of a specific permutation
QUESTION [9 upvotes]: Let $n = 2k \gt 0$ be an even integer, and $S_n$ the symmetric group on $\{1,..,n\}$.
Let $\mu$ be the permutation s.t. $\mu(x) = n-x+1$.
Question: What is the centralizer of $\mu$ in $S_n$ ?  That is, what is the subgroup $Z(\mu)$ of all permutations in $S_n$ that commute with $\mu$ ?
Note that $\mu$ is the product of $k$ disjoint transpositions, and thus an involution (self-inverse), $\mu = (1\;n) (2\;n-1) \ldots (k\;k+1)$.  Because they are disjoint, these transpositions commute, and it follows that $\mu$ commutes with any product of a subset of these.
But such products do not generally account for the entire centralizer $Z(\mu)$.  For example with $n=6$, the permutation $(1\;3) (4\;6)$ commutes with $\mu$ but is not expressible as a product of the disjoint transpositions above that form $\mu$.
If possible an explicit characterization of $Z(\mu)$ elements would be preferred, but a construction of generators (valid for all $n$) would also be useful.  The application I have in mind is the enumeration of diagonal latin squares; see also a related question I posted previously.
The idiosyncratic term used there, $\mu$-derangement, may be justified by its definition being equivalent to both a permutation $g$ and $\mu g$ being derangements (permutations without fixed points).  Computational experience suggests that $\mu$-derangements which share certain easily computed invariants result in equal numbers of "normalized" latin squares, and that the broadest extent of such equivalence may derive from conjugation by permutations in $Z(\mu)$.

REPLY [5 votes]: The centralizer of $\mu$ is a example of a wreath product: Specifically, it is isomorphic to $C_{2} \wr S_{k},$ so its order is indeed $k! 2^{k}.$ The "base group" is elementary Abelian group of order $2^{k},$ and is generated by the $k$-disjoint transpositions appearing in $\mu,$ and the $S_{k}$ permutes these $k$ generating transpositions around as if they were individual points. But it should be noted that the centralizer is definitely not a direct product- it is a particular kind of semi-direct product, but the $S_{k}$ induces non-trivial automorphisms of the base group.