Find all the elements of order $63$ in $S_{50}$

Question

Find all the elements of order 63 in the permutation group $S_{50}$. I know that the number of elements of $S_{50}$ is $50!$, and $63=9\cdot7$, and that the number of cycles is $62!$, but I don`t quite know how to provide an answer. I am new to this type of problems and I do not have many examples, could you provide a full proof, or at least in the form of an answer, such that it would serve as a model for similar problems I encounter? Thank you very much!!!

Answer 1

One of the cycle types of order $63$ in $S_{50}$ consists of a $7$-cycle and a $9$-cycle.

For the $7$-cycle, there are $50$ possibilities for the first entry, $49$ for the second, $48$ for the third, $47$ for the fourth, $46$ for the fifth, $45$ for the sixth, and $44$ for the seventh; that's $50\times49\times48\times47\times46\times45\times44$ possibilities. However, since $(abcdefg)=(bcdefga)=(cdefgab)=(defgabc)=(efgabcd)=(fgabcde)=(gabcdef)$, we divide by $7$, to get $71916768000$ possibilities.

Likewise, for the $9$-cycle, there are $43\times42\times41\times40\times39\times38\times37\times36\times35/9=22737334838400$ possibilities.

Overall, the number of possibilities for a $7$-cycle and a $9$-cycle is the product of those two large numbers.

You would also have to do similar computations to get the other possibilities with other cycle types.

Answer 2

I am afraid this is not a complete answer, but the number of elements of order $63$ in $S_{50}$ is — wait for it —

$$887785975527743696226835481189442937805320172949463756800000$$

which is about $8.88 \times 10^{59}$, so I am afraid that the known universe lacks the resources to list them all.

These elements fall into $96$ different cycle types, so even writing them down would be somewhat tedious.