Let $S_n$ be the symmetric group of order $n$ with $g_1,s\in S_{n-1}$ and $i,j\in\{1,2,\ldots,n\}$. Then, what is the general structure of elements $s$ such that the equation $$g_1s(1,2,n)g_1^{-1}=(i,j,n)$$ So, basically, it is like asking which elements when multiplied by $(1,2,n)$ to the right, are conjugate to $(i,j,n)$.
I know that elements having the same cycle structure are conjugate to each other. But, how do we determine that $s(1,2,n)$ will have the same structure. One trivial example for $s$ is the identity. But, are there other elements for $s$ as examples? Thanks beforehand.