I'm trying to solve this problem from my abstract algebra course:
Find a subgroup of $S_5$ (Symmetric group of order 5) isomorphic to the quaternion group $Q$.
I started writting down the elements of $Q$ to start trying some examples and see if the quaternion properties where verified: $$Q=\{\pm 1,\pm i,\pm j,\pm j\}.$$ Obviously the element $1$ is $(1)(2)(3)(4)(5)$ in $S_5$.
Then I tried with: $$i=(1234)(5)\ \ , \ \ -i=(1432)(5).$$
Both verify they have order $5$, and from them I get $-1=(13)(24)(5)$. Now, I'm stuck, since I think I pick wrong elements because I can't find good elements for $\pm j$ and $\pm k$. What is the easiest way to do this kind of problem where you're asked to find a subgroup isomorphic to a certain group?
Any help will be appreciated.