We assign to each positive integer n a digraph Γ(n, 11) whose set of vertices is Zn = {0, 1, 2, ..., n − 1} and there exists exactly one directed edge from a to b if and only if a11 ≡ b(mod n), where a, b ∈ Zn. Let Γ1(n, 11) be the subdigraph induced by the vertices which are coprime to n. We discuss when the subdigraph Γ1(n, 11) is regular or semi-regular. A formula for the number of fixed points of Γ(n, 11) is established. A necessary and sufficient condition for the symmetry of the digraph Γ(n, 11) is proved. Moreover, using Carmichael ́s lambda function, the number of components and conditions for the existence of cycles in the digraph Γ(n, 11) is presented.