I am given that G is a set of elements that commute with (12)(3456) in $S_6$, and asked following questions: i) Show that $G$ is a subgroup of $S_6$.
Attempt( I could just do the first subgroup test, which was trivial): a) So, $e$ commutes with (12)(3456) and thus $e$ $\in$ G, hence G is not empty.
The reason I could not proceed now is because I do not yet know what other elements belong G. Without knowing it, I do not know how to show that it is closed and inverse exists.
Question: How do I know what elements in $S_6$ commute with (12)(3456). Is this question the same as find the elements in $S_6$ that are conjugate with (12)(3456)? I am pretty sure that there is a neat idea to proceed instead of trying every possible combination, but I just do not see it.