Let $G$ be a non-abelian finite group. Let $C_1$ and $C_2$ be distinct conjugacy classes in $G$ with following conditions:
1) $C_1$ and $C_2$ contain of elements of same prime order $p$.
2) For every complex irreducible character of $G$, its values on $C_1$ and $C_2$ are complex conjugates (in case they are real, they should be same).
Can we conclude that the cyclic subgroup generated by any element of $C_1$ and one such by any element of $C_2$ are conjugate in $G$? (If yes, how? and if not, what is example?)
In the book on Finite Simple Groups (First proceeding) an article mentions such fact concerning $G$ to be a Janko group and $C_1$ and $C_2$ certain conjugacy classes of elements of same prime order.