Example 4-7 A circular ring, with beads of red or blue installed at the positions of $0^{\circ}, 90^{\circ}, 180^{\circ}, 270^{\circ}$ in a clockwise direction, how many different equivalence classes are there? That is, how many different schemes are there? Schemes that can be made to coincide by rigid body motion are considered the same.

This problem can also be viewed as the case of the four squares in the previous example, as shown in Figure 4-8, but the square is a transparent glass plate, colored with red and blue. How many coloring schemes are there?
For the higher permutation group, in addition to $p_{1}, p_{2}, p_{3}, p_{4}$ as in the previous example, it also includes:
(5) A $180^{\circ}$ flip along the $x^{\prime} x$ axis, corresponding to the permutation:
$$
\begin{array}{l}
p_{5}=\left(c_{1}\right)\left(c_{2}\right)\left(c_{3} c_{6}\right)\left(c_{4} c_{5}\right)\left(c_{7} c_{10}\right)\left(c_{8} c_{9}\right)\left(c_{11} c_{12}\right)\left(c_{13}\right)\left(c_{15}\right)\left(c_{14} c_{16}\right) \\
c_{1}\left(p_{5}\right)=4
\end{array}
$$
(6) A $180^{\circ}$ flip along the $y^{\prime} y$ axis, corresponding to the permutation:
$$
\begin{array}{l}
p_{6}=\left(c_{1}\right)\left(c_{2}\right)\left(c_{3} c_{4}\right)\left(c_{5} c_{6}\right)\left(c_{7} c_{8}\right)\left(c_{9} c_{10}\right)\left(c_{11} c_{12}\right)\left(c_{13} c_{15}\right)\left(c_{14}\right)\left(c_{16}\right) \\
c_{1}\left(p_{6}\right)=4
\end{array}
$$
(7) A $180^{\circ}$ flip along the diagonal 13, corresponding to the permutation:
$$
\begin{array}{l}
p_{7}=\left(c_{1}\right)\left(c_{2}\right)\left(c_{3}\right)\left(c_{4} c_{6}\right)\left(c_{5}\right)\left(c_{7}\right)\left(c_{8} c_{10}\right)\left(c_{9}\right)\left(c_{11}\right)\left(c_{12}\right)\left(c_{13} c_{14}\right)\left(c_{15}\right)\left(c_{16}\right), \\
c_{1}\left(p_{7}\right)=8
\end{array}
$$
(8) A $180^{\circ}$ flip along the diagonal 24, corresponding to the permutation:
$$
\begin{array}{l}
p_{8}=\left(c_{1}\right)\left(c_{2}\right)\left(c_{3} c_{5}\right)\left(c_{4}\right)\left(c_{6}\right)\left(c_{7} c_{9}\right)\left(c_{8}\right)\left(c_{10}\right)\left(c_{11}\right)\left(c_{12}\right)\left(c_{13} c_{16}\right)\left(c_{14} c_{15}\right) \\
c_{1}\left(p_{8}\right)=8
\end{array}
$$

According to Burnside's formula, the number of different equivalence classes is:
$$
l=\frac{1}{8} \times[16+2+4+2+4+4+8+8]=\frac{1}{8} \times[48]=6
$$
The 6 different schemes are consistent with Figure 4-7, but their meanings are not entirely the same, including various types of flips. Please understand them on your own.