Recently I have been studying the quaternions, generalized quaternions, and dicyclic groups. Initially, I was interested in finding the minimum degree permutation representations of such groups and was happy to find many resources that helped me.
However, now I am interested in finding 'larger' degree representations. I know that for any finite group $G$ that: $$ G\hookrightarrow{}S_{|G|} $$ However, I wish to explore faithful permutation representations of these groups that can be embedded into a symmetric group larger than $S_{|G|}$. Is this a fruitful avenue to explore, and if so, what are the key terms that I should be searching for?
We have $S_m\hookrightarrow S_n$, for $n\ge m$, in a natural way (leave $\{m+1,\dots,n\}$ fixed, say).
As far as the embedding, you could just compose the above with the Cayley embedding, $G\hookrightarrow S_{|G|}$, to get one into $S_n$ for any $n\ge |G|$.
So the answer to your question is yes, and this amounts to a fairly trivial add-on to the Cayley theorem.