I know that if $|G| = n$, then $G$ can be embedded into $S_n$. But the group $S_n$ is very large compared to $G$, so I was wondering if there are general ways of embedding $G$ into a smaller symmetric group. (By general I don't mean that it has to work for all groups, but hopefully for large classes of groups)
Also, I was wondering embedding groups into smaller symmetric groups is common/useful, or just a curiosity?
The only approach I could think of is to let $G$ act on various things, and hope that the action is faithful. One nice thing, for example, is that if $G$ has a simple subgroup $H$ (which is itself non-normal in $G$), then the action of $G$ on the left cosets of $H$ is faithful, since according to this question, the kernel must be trivial.
Furthermore, if you let $G$ act by conjugation on a subgroup, I believe the action is sometimes faithful, sometimes not (has to do with weather the conjugates are disjoint).