Find the number of elements in $S_6$ conjugate to $(123)(456)$
I know we're only looking at elements in $S_6$ with the same cycle type as $(123)(456)$ (two 3-cycles).
So we have the following:
$$\frac{6\cdot 5 \cdot 4}{3}\cdot \frac{3\cdot 2 \cdot 1}{3}\cdot \frac{1}{2}=40$$
But this question has been asked before and there was a different answer: How do you calculate the number of elements conjugate to a particular permutation?
I know mine is right but why is the other wrong? Is it because we are totally disregarding order, i.e. we're acting as if $(123)=(213)$?