Example 7.8 Now we use red, blue, and yellow to color the 12 edges of a cube $V$, such that the number of edges colored red, blue, and yellow are $3, 3, 6$, respectively. Find the number of distinct edge-coloring patterns of the cube.
Let $(D, *)$ be the rotation group of a cube $V$, and $(W, \circ)$ be the edge permutation group induced by $(D, *)$. Suppose $d \in D$, and the edge permutation induced by $d$ is $w$. Let $V_{1}$ be a cube obtained by coloring the 12 edges of $V$, where the number of red, blue, and yellow edges in $V_{1}$ are $3, 3, 6$, respectively. Then $d V_{1}=V_{1}$ if and only if the edges in the same cycle of $w$ have the same color. The cycle index of $(W, \circ)$ is
$$
P_{W}\left(x_{1}, x_{2}, \cdots, x_{12}\right)=\frac{1}{24}\left(x_{1}^{12}+8 x_{3}^{4}+6 x_{1}^{2} x_{2}^{5}+3 x_{2}^{6}+6 x_{4}^{3}\right).
$$

For the edge permutation of type $1^{12}$, the number of coloring methods such that the edges in the same cycle have the same color is $\frac{12!}{3!3!6!}=18480$;

For the edge permutation of type $3^{4}$, the number of coloring methods such that the edges in the same cycle have the same color is $\frac{4!}{1!1!2!}=12$;

For the edge permutation of type $1^{2} 2^{5}$, the number of coloring methods such that the edges in the same cycle have the same color is $2 \cdot \frac{5!}{1!1!3!}=40$;

For the edge permutations of type $2^{6}$ and type $4^{3}$, there are no coloring methods such that the edges in the same cycle have the same color.

Thus, by Burnside's Lemma, the number of distinct edge-colored cubes is
$$
N=\frac{1}{24}(18480+8 \times 12+6 \times 40)=784.
$$

Let $A_{n}$ be a regular $n$-gon, and $(C, *)$ and $(D, *)$ be the normal rotation group and the improper rotation group of $A_{n}$, respectively. Suppose we color the $n$ vertices (or $n$ edges) of $A_{n}$ using $m$ colors, with each vertex (or each edge) colored with one color. Let the set of all such vertex (or edge) colored $n$-gons be denoted by $X$. Define a mapping $f$ from $C \times X$ (or $D \times X$) to $X$ as follows: If $g \in C$ (or $g \in D$) and $x \in X$, and the vertex (or edge) colored $n$-gon obtained by applying the rotation $g$ to $x$ is $x^{\prime}$, then $f((g, x))=x^{\prime}$. The group $(C, *)$ or $(D, *)$ acts on the set $X$ through the mapping $f$. The number of $C$-orbits and $D$-orbits on $X$ are called the number of type I patterns and type II patterns on $X$, respectively.