Example 7.5 Let $\left(D^{\prime}, \circ\right)$ be the vertex permutation group derived from the improper rotation group of a regular hexagon. Find the cycle index of $\left(D^{\prime}, \circ\right)$.
Solution: By Theorem 7.15 and Example $7.4$, the cycle index of $\left(D^{\prime}, \circ\right)$ is
$$
\begin{aligned}
P_{D^{\prime}} & \left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}\right) \\
& =\frac{1}{2 \times 6} \sum_{d \mid 6} \varphi(d)\left(x_{d}\right)^{\frac{6}{d}}+\frac{1}{4}\left(x_{2}^{3}+x_{1}^{2} x_{2}^{2}\right) \\
& =\frac{1}{12}\left(x_{1}^{6}+x_{2}^{3}+2 x_{3}^{2}+2 x_{6}\right)+\frac{1}{4}\left(x_{2}^{3}+x_{1}^{2} x_{2}^{2}\right) \\
& =\frac{1}{12}\left(x_{1}^{6}+4 x_{2}^{3}+2 x_{3}^{2}+3 x_{1}^{2} x_{2}^{2}+2 x_{6}\right) .
\end{aligned}
$$