How to find the number of left coset of $H$ in symmetric group $S_7$

Question

It is given that $\sigma=\,(124)(3567)$, let $H$ be the subgroup of symmetric group $S_7$ generated by $\sigma$.

Find the number of left coset of $H$ in symmetric group $S_7$.

My attempt: I tried to find the order of $\sigma$, then we can use the Langrange Theorem to find the number of left coset of $H$ in symmetric group $S_7$. However, finding the order of $\sigma$ is very tedious, since $\sigma^{8}$ is still not equals to the identity permutation.

Could anyone give me a method or a hint?

Answer 1

Hint: Because the two cycles are disjoint, for every integer $k$ $$\sigma^k=(124)^k(3\ 5\ 6\ 7)^k.$$