In the $S_7$ group, in action on itself by $xy=xyx^{-1}$, I would like to calculate the orbit of $(123)(456)$.
I read the definition:
Consider a group $G$ acting on a set $X$. The orbit of an element $x$ in $X$ is the set of elements in $X$ to which $x$ can be moved by the elements of $G$. The orbit of $x$ is denoted by $G\cdot x$: $G\cdot x = \{g\cdot x | g\in G\}$
But I can't seem to understand how to calculate technically speaking the orbit of $(123)(456)$. My goal is to calculate $|Stab_{S_7}(123)(456)|$. In order to achieve it I have to calculate the orbit, but how?
If you have a permutation written down as its cycle decomposition, it's very easy to describe the action of an element by conjugation - that element is applied to all of the entries in the cycles. If $y$ includes the cycle $(a_1a_2\dots a_k)$, $\sigma y\sigma^{-1}$ includes the cycle $(\sigma(a_1)\sigma(a_2)\dots\sigma(a_k))$.
So then, we start with an element that's a composition of two disjoint $3$-cycles. Conjugate by something, and we get a composition of two disjoint $3$-cycles. Can you see what the orbit will be now?