Example 4-13 The 6 faces of a cube are to be colored using red and blue. How many different coloring schemes are there?
The rigid motion group that makes a regular hexahedron coincide has the following cases:
(1) Identity permutation, i.e., the unit element (1)(2)(3)(4)(5)(6), formatted as $(1)^{6}$.
(2) Rotation by $\pm 90^{\circ}$ around the axis $A B$ passing through the centers of faces (1) and (6) (as shown in Figure 4-15(a)). The corresponding permutations are
(1)(2345)(6), (1)(5432)(6)

formatted as $(1)^{2}(4)^{1}$. Since a regular hexahedron has 3 pairs of opposite faces, there are 6 permutations of this type.
(3) Rotation by $180^{\circ}$ around the axis $A B$ results in (1)(24)(35)(6). The format is $(1)^{2}(2)^{2}$, and there are 3 permutations of this type.
(4) Rotation by $180^{\circ}$ around the axis $C D$ (as shown in Figure 4-15(b)) results in the permutation (16)(25)(34), formatted as $(2)^{3}$.

In a regular hexahedron, there are 6 pairs of parallel edges in diagonal positions, so there are 6 permutations of this type.
(5) Rotation by $\pm 120^{\circ}$ around the diagonal $E F$ (as shown in Figure 4-15(c)). Rotation by $120^{\circ}$ around $E F$ results in the permutation $(346)(152)$. Rotation by $-120^{\circ}$ around $E F$ results in the permutation $(643)(251)$, formatted as $(3)^{2}$.
Since a regular hexahedron has 4 diagonals, there are 8 permutations of this type.
Thus, the number of different coloring schemes is
$$
\begin{aligned}
M & =\frac{1}{24} \times\left[2^{6}+6 \cdot 2^{3}+3 \cdot 2^{1}+6 \cdot 2^{3}+8 \cdot 2^{2}\right] \\
& =\frac{1}{24} \times[64+48+48+48+32]=10
\end{aligned}
$$