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I know that the subgroups of $S_3$ are $\{e\}$, $\langle(12)\rangle$, $\langle(13)\rangle$, $\langle(23)\rangle$, $A_3$, and $S_3$. What I also know is that conjugation is $C_g(H) = gHg^{-1}$. Thus in order to compute $S_3$ acting by conjugation on $X$ would I do this just by taking one of the subgroups and then setting $g$ equal to one of the other subsets?

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If you want to find the orbits of this action, then you take an element $H$ of $X$ (that is, some subgroup of $S_3$) and then compute $gHg^{-1}$ for every $g\in S_3$. Next, take another $H\in X$, one that is not in the orbit you just found, and do the same.

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