Exercise 4.3.12 (Representatives for conjugacy classes of elements of order $4$ in $S_8$ and in $S_{12}$)

Proof. Consider a permutation $\sigma \in S_n$ and its cycle decomposition. Then the order of $\sigma$ is the least common multiple of the orders of the cycle of the decomposition. If $|\sigma |=4$ then all non trivial cycles have order $2$ or $4$, thus $|\sigma |=4$ if and only all cycles have order $2$ or $4$, and one of them has order $4$.

• If $n=8$, the representative for each conjugacy class of elements of order $4$ in $S_8$ are

  $$\begin{array}{llll}\hfill & (1234),& \hfill & \\ \hfill & (1234)(56),& \hfill & \\ \hfill & (1234)(56)(78),& \hfill & \\ \hfill & (1234)(5678).& \hfill & \end{array}$$

• If $n=12$, there are at most $3$ cycles of order $4$, so we obtain

  $$\begin{array}{llll}\hfill & (1234),& \hfill & \\ \hfill & (1234)(56),& \hfill & \\ \hfill & (1234)(56)(78),& \hfill & \\ \hfill & (1234)(56)(78)(910),& \hfill & \\ \hfill & (1234)(56)(78)(910)(1112),& \hfill & \\ \hfill & (1234)(5678),& \hfill & \\ \hfill & (1234)(5678)(910),& \hfill & \\ \hfill & (1234)(5678)(910)(1112),& \hfill & \\ \hfill & (1234)(5678)(9101112)& \hfill & \\ \hfill & & \hfill \end{array}$$