conjugacy in symmetric group I was reading Dummit and Foote and encountered the following statement: any two elements in $S_n$ are conjugate if and only if they have the same cycle types.
However, I am able to produce a counter example:
Let $(1 2 3)$ and $(4 5 6) (7 8)$ be in $S_{10}$, then $(4 5 6) (7 8) =(1 3 2) (4 5 6) (7 8) (1 2 3)$, which show that these two are conjugate.
What am I misunderstanding here?

I think that you believe you found a counter example because you think that
$(456)(78)$ and $(132)(456)(78)(123)$ are two different cycle types of the same element.
This is not true since $(132)(456)(78)(123)$ is not a product of disjoint cycles.
(Maybe I have misunderstood your misunderstanding)
