The easy way: Are you familiar with the orbit-stabiliser theorem? If so, can you think up a way of using this? Which elements commute with your n-cycle?

The harder way: Alternatively, note that when n is odd, conjugation of a long-cycle by (12) must be "undone" by (12). There is no way to get from

without conjugation by (12) when

is odd.

(When dealing with these kinds of problems it is perhaps helpful to remember that

- every coefficient is acted on individually by the permutation you are conjugating by).