Let $G$ be a finite group of order $n$ generated by permutations $g_1,\dots,g_s$ such that $g_1g_2\cdots g_s$ is the identity permutation.
Consider the permutation representation obtained by embedding $G$ into $S_n$. Can we relate the character of the permutation representation of $G$ on the cosets $G/\langle g_i\rangle$ to the number of cycles of $g_i$?
The character of the permutation representation of $G$ on the cosets $G/\langle g_i\rangle$ equals the number of points of $G/\langle g_i\rangle$ fixed by $g_i$, but I'm not sure how to think about this. I believe that the quotient makes it different from just the number of 1-cycles that $g_i$ has, but I'm not sure how to go about computing this character.
I am also wondering if there are embeddings that this might be easier for. Is there a permutation representation of $G$ that we can choose so that there is a "nice" relationship between the character of the permutation representation of $G$ on the cosets $G/\langle g_i\rangle$ to the number of cycles of $g_i$?