Example 2 As shown in Figure 1-10, 5 beads are strung together to form a necklace. Now, there are 3 colors of beads available for use. How many different necklaces can be made?


Solve $N_{5}=\{1,2,3,4,5\}, m=3, N_{5}$ on the permutation group $G=\left\{P_{1}=I, P_{2}, P_{3}, \cdots, P_{10}\right\}$, where $P_{1}, P_{2}, \cdots, P_{5}$ are the same as in Example 1, and
$P_{6}=(1)(25)(34)$, the reflection about the line through the center and 1,
$P_{7}=(2)(1,3)(4,5)$, the reflection about the line through the center and 2,
$P_{8}=(3)(2,4)(15)$, the reflection about the line through the center and 3,
$P_{9}=(4)(35)(12)$, the reflection about the line through the center and 4,
$P_{10}=(5)(14)(23)$, the reflection about the line through the center and 5.

Thus, by Theorem 1, the number of different necklaces is
$$
M=\frac{1}{10}\left(3^{5}+3+3+3+3+3^{3}+3^{3}+3^{3}+3^{3}+3^{3}\right)=39 .
$$