Commutativity of cycles in decomposition of commutative permutations

Question

Let $\sigma,\tau \in{S_n}$ such that $ \sigma \tau = \tau \sigma $. Let $\sigma = c_1c_2..c_i, \tau = C_1C_2...C_j$ be the decompositions in disjoint cycles of $\sigma, \tau$. Then, for any $k,l$, do $c_k$ and $C_l$ commute?