Smallest $n$ such that $S_n$ contains a subgroup isomorphic to $C_5$? The Cayley's theorem tells us that every finite group is isomorphic to a subgroup of some $S_n$
I have no idea how to go on about this question. I can list every elements of $S_1 , S_2, S_3, ... $ but I'm guessing that's not how it wants me to do it. I also have to find it for $C_2 \times C_2 \times C_2$ and $S_3 \times S_3$ but could anyone please explain the concept using the $C_5$ example? And maybe I will be able to try the rest myself.
Thank you.

Hint:
An element of $\;S_n\;$ has order $\;p\,,\,\,p\;$ a prime, iff it is a product of disjoint $\;p$-cycles, for example $\;(123)\;,\;\;(123)(345)\;$ are elements of order 3 in $\;S_6\;$ .
