# GCD and the cycle decomposition of a permutation

Take a permutation $\sigma \in \mathcal{S}_n$. Its cycle decomposition is the (essentially) unique decomposition in disjoint cycles : $\sigma = c_1 c_2 \cdots c_k$. Write $p_i$ the length of each cycle $c_i$. The lowest common multiple of all the $p_i$ is the order of the permutation, but is there anything interesting about the greatest common divisor of those same $p_i$ ? $$GCD(p_1, \cdots p_k)\ =\ ?$$ Notably, is there a name for permutations verifying $GCD(p_1, \cdots p_k) = 1$

Thanks,