Find a subgroup of order $120$ in $S_8$

Question

Find a subgroup of order $120$ in $S_8$

Listing the possible $k$-cycles in $S_8$. I have that the possible orders of the elements in this group are $1, 2, 3, 4, 5, 6, 7, 8, 10, 12$ and $15$. How can I get a subgroup with such an order?

Answer 1

We know that $120=5!.$ Can you find an $S_5$ in $S_8?$

Answer 2

The underlying set of the group $S_8$ is the set of bijections from $N=\{1,2,3,4,5,6,7, 8\}$ to $N$. Can you show that the underlying set of $S_5$ is in bijection with the set of bijections just mentioned that happen to fix $6,7,8$? Can you show that such a set is a subgroup of $S_8$? What is the order of $S_5$?

Answer 3

Hint: There are clearly at least $8 \choose 3$ copies of $S_5$ in $S_8$.