The question is: Find an element $τ$ in S6 such that $τ$ is a 6-cycle (i.e., $τ=(a,b,c,d,e,f)$ in cyclic notation, where $a,\dots,f$ belong to $\{1,...,6\}$), $τ$ commutes with $σ$ (i.e. $στ=τσ$). $σ$ is defined by $σ(i) = i + 3$ for $i <= 3$ and $σ(i) = i - 4$ for $i >= 4$.
So far my solution is that $τ$ is just simply $τ = (1, 2, 3, 4, 5, 6)$ and that it solves for all $(τσ)(i) = (τ(σ(i))$. I hope that I'm not too far off.