same cycle type but not conjugate

Question

What's an example of $\alpha,\beta$ in the alternating group $A_n$ for some $n\geq 4$, such that $\alpha,\beta$ have the same cycle type but not conjugate in $A_n$?

Answer 1

For example in $A_4$, the alternating group on 4 symbols, consisting of the even permutations in $S_4$. The elements $(1\ 2\ 3)$ and $(1\ 3\ 2)$ of $A_4$ have the same cycle structure, but they are not conjugate in $A_4$. That is, there are elements $g$ in $S_4$ such that $g^{-1}(1\ 2\ 3)g=(1\ 3\ 2)$, but there is no such element in $A_4$.

Reference: Why are two permutations conjugate iff they have the same cycle structure?