Determining the conjugacy classes of a wreath product $G \wr S_n$

Question

If $G$ is a finite group and its conjugacy classes are known, can the conjugacy classes of the wreath product $G \wr S_n \cong G^n \rtimes S_n$ be determined?

Answer 1

This is indeed handled in Section 4.2 of James and Kerber's book on Representations of the Symmetric group.

We also considered this problem in Section 2 our paper describing a computer algorithm for computing conjugacy class representatives in permutation groups.

J. Cannon and D. Holt, Computing conjugacy class representatives in permutation groups, J. Algebra 300 (2006), 213--222.

I can cut and paste the main bit of theory you need. The problem easily reduces to finding those classes that map onto a specific element of $S_n$.

Let $A$ be any group, let $P$ be a permutation group acting on the set $\{ 1\ldots d\}$, and let $W := A \wr P$. Then $W$ is a semidirect product of $A^d$ by $P$, and elements of $W$ have the form $(g,x)$ with $g \in P$, $x \in A^d$, and $x = (x_1,\ldots,x_d)$ with $x_i \in A$. In general, for $x \in A^d$, we denote the $i$-th component of $x$ by $x_i$. The action of $P$ on $A^d$ in $W$ is given by: $$(g,1)^{-1}(1,(x_1,\ldots,x_d))(g,1) = (1,(x_{1^{g^{-1}}},\ldots, x_{d^{g^{-1}}})).$$

Hence, for $g \in P$, $x,z\in A^d$, we have \begin{equation}\label{conjeqn} (1,z)^{-1}(g,x)(1,z) = (g,y), \ \mathrm{where}\ y_i = z^{-1}_{i^{g^{-1}}}x_iz_i \ \mathrm{for}\ 1 \le i \le d. \end{equation}

$\textbf{Theorem}$ With the above notation, let $x^{(1)},\ldots,x^{(k)}$ be representatives of the conjugacy classes of $A$. Fix an element $g \in P$, and let $r_1,r_2,\ldots,r_s$ be representatives of the cycles of $g$ in its action on $\{ 1\ldots d\}$. Then $$ R := \{\,(g,x) \mid x_i = 1 \ \mathrm{for}\ i \not\in \{r_1,\ldots,r_s\},\, x_i \in \{x^{(1)},\ldots,x^{(k)}\}\ \mathrm{for}\ i \in \{r_1,\ldots,r_s\}\,\} $$ is a set of representatives of the $A^d$-classes of elements of $W$ of the form $(g,x)$ for $x \in A^d$.

The size of the $A^d$-class $R_{gx}$ containing $(g,x)$ is the product of the sizes of the classes of the $x_i$ in $A$ with $i \in \{r_1,\ldots,r_d\}$ with the lengths of the cycles of $g$ that contain $r_1,\ldots,r_d$.

The above describes the $A^d$-classes of $W$; i.e. the orbits under conjugation by the subgroup $A^d$ of $W$. To get the classes of $W$ itself, first of all we only consider those $R_{gx}$ as $g$ ranges over a set of class representatives of $P$. Then, for each such $g \in W$, the $A^d$-class representatives $R_{gx}$ are fused under the action of $C_P(g)$. This action is the same as the action of $C_P(g)$ on the cycles of $g$, so is easy to calculate.

Then the size of a $W$-class $R_{gx}$ is the product of the size of the class of $g$ in $P$ with the sum of the the sizes of the $A^d$-classes $R_{gx'}$ that lie in the orbit under the action of $C_P(g)$.