Finding the conjugacy classes of $S_{3}$ I'm having difficulty in understanding how the conjugacy classes are calculated in $S_{3}$.
I know they are $\{e\}, \{(12),(13),(23)\}$ and $\{(123),(132)\}$ but do not know how to show for the latter two.
For $(12)$ in $S_{3}$, to find the conjugacy class, I know any conjugate will have
the form $(g(1) g(2))$ where $g$ is the conjugating element. But the notes I have go on to say this: $(12)^{(23)} = ((23) · 1 (23) · 2) = (13)$ and $(12)^{(123)} = (23)$.
I can't understand why these calculations are done and what they mean and wondered if anyone could explain what is happening in this line

I'm not sure what the caclulations are done, but the notation $(12)^{(23)}$ means that $(12)$ is conjugated by $(23)$, i.e. $(12)^{(23)} = (23)(12)(23)^{-1}$. I guess you can perform the composition of the cycles now and determine the value. 
On the other hand in $S_n$ we have that two elements are conjugates iff they have the same kind of disjoint cycles. This means that the elements of the form $(a_1,a_2)$ are in one conjugacy class, while $(a_1,a_2,a_3)$ are in other.
