For each positive integer $n$, write $S_n$ for the symmetric on $n$-letters. Suppose that $m | n$ is a proper divisor of $n$, and write $n = km$. Consider the element

$$\displaystyle u(m,n) = \underbrace{\begin{pmatrix} 1 & 2 & \cdots & k\end{pmatrix} \cdots \begin{pmatrix} n-k+1 & n-k+2 & \cdots & n \end{pmatrix}}_m$$

Let $C(u)$ be the centralizer of $u$ in $S_n$. The orbit of $u$ under conjugation by $S_n$ is the set of elements in $S_n$ with the same cycle-type as $u$, and the size of the orbit is readily computed to be

$$\displaystyle \binom{km}{k} (k-1)! \binom{k(m-1)}{k} (k-1)! \cdots \binom{2k}{k} (k-1)! (k-1)! \frac{1}{m!} = \frac{n!}{k^m \cdot m!},$$

so by the orbit-stabilizer theorem, it follows that $C(u)$ is a group of cardinality $k^m \cdot m!$.

Is there a name for these groups? Did anyone study them in particular in the literature? I am interested in knowing when such groups (or their subgroups) can be realized as the Galois group of a polynomial of degree $n$.

For example, Lehmer's polynomial

$$\displaystyle x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$$

has Galois group order $1920$ (see Is Lehmer's polynomial solvable?), and the corresponding value for the size of $C(u)$ in this case is $2^5 \cdot 5! = 3840$. It can be shown that the Galois group isomorphic to a subgroup of $C(u)$ (and in fact can be embedded by some natural ordering).