Example 7.4 Let $(G, \circ)$ be the vertex permutation group derived from the normal rotation group of a regular hexagon. Find the cycle index of $(G, \circ)$.
$$
\begin{array}{l}
P_{G}\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}\right) \\
=\frac{1}{6} \sum_{d \mid 6}^{1} \varphi(d)\left(x_{d}\right)^{\frac{6}{d}} \\
=\frac{1}{6}\left[\varphi(1) x_{1}^{6}+\varphi(2) x_{2}^{3}+\varphi(3) x_{3}^{2}+\varphi(6) x_{6}\right] \\
=\frac{1}{6}\left(x_{1}^{6}+x_{2}^{3}+2 x_{3}^{2}+2 x_{6}\right) .
\end{array}
$$

Let $A_{n}, A_{n}(k)(k=0,1,2, \cdots, n-1), c_{k}(k=0,1,2, \cdots, n-1)$ have the same meanings as before. Label the $n$ edges of $A_{n}$ sequentially (in a counterclockwise direction) with the numbers $1,2, \cdots, n$, and denote the edge labeled $i(i=1,2, \cdots, n)$ as $e_{i}$. Let $k$ be any non-negative integer not greater than $n-1$, and let $s=n-k$. Suppose $e_{i}(1 \leqslant i \leqslant n)$ is any edge of $A_{n}$, then when $i \leqslant s$, the edge $e_{i}$ of $A_{n}$ coincides with the edge $e_{i+k}$ of $A_{n}(k)$; when $i>s$, the edge $e_{i}$ of $A_{n}$ coincides with the edge $e_{i+k-n}$ of $A_{n}(k)$. Therefore, $c_{k}$ corresponds to the following permutation on the edge set $E=\left\{e_{1}, e_{2}, \cdots, e_{n}\right\}$ of $A_{n}$:
$g_{k}^{\prime}$ is called the edge permutation induced by $c_{k}$.
Let $G^{\prime}=\left\{g_{0}^{\prime}, g_{1}^{\prime}, g_{2}^{\prime}, \cdots, g_{n-1}^{\prime}\right\}$. Similar to the vertex permutation group $(G, \circ)$ induced by the normal rotation group of a regular $n$-gon, it is easy to prove that $\left(G^{\prime},{ }^{\circ}\right)$ is an $n$-order cyclic group with $g_{0}^{\prime}$ as the identity element and $g_{1}^{\prime}$ as the generator, where the operation $\circ$ is the usual permutation multiplication. The group $\left(G^{\prime}, \circ\right)$ is called the edge permutation group induced by the normal rotation group of a regular $n$-gon.