I found the following question in my lecture notes:
Consider the following subgroup of $S_{8}:\ \langle(1 5 7 2 4 8), (1 4 7)(2 5 8)(3 6)\rangle$. Decompose this subgroup as the direct product of cyclic subgroups.
I was hoping someone could guide me through the process of finding the direct product of the cyclic subgroups. I am not interested in "the answer", but in the intuition that's needed to solve the question. Thanks in advance!