is there a way to describe the convex hull of the set of permutation matrices with exactly one non-trivial cycle?

or maybe I should ask for the convex hull of cycle matrices :

let $(i_{1},..,i_{k})$ be a cycle then $A$ is a cycle matrix if the entries $(i_{1},i_{2})$ ...$(i_{k},i_{1})$ are $1/k$, and all the others are 0. (it seems to me that this convex hull is the set of non-negative matrices of entries sum equal to 1)