I am reading Dummit and Foote's proof that $A_5$ is simple, and I saw the statement
$$\text{All twenty 3-cycles are conjugate in $A_5$}$$
However, I tried to conjugate $(123)$ into $(132)$, and I couldn't find an even conjugator. I believe the way to find all conjugators is as follows: $$ \begin{bmatrix} (123) & (123) & (123)\\ (132) & (321) & (213) \end{bmatrix} $$ By reading vertically, we see that the possible conjugators are $(23)$, $(13)$, $(12)$, all of which are odd.
I think I must be wrong, there must be other conjugators; but on the other hand I don't see how there could be any others. What is going on here? Thank you very much.