We study a family of vertex transitive digraphs whose vertices represent the k-permutations of n elements. After showing some general properties, we concentrate upon the study of the symmetry of these digraphs. By using some distance-related properties, their automorphism groups are characterized. We also characterize those digraphs which are Cayley digraphs. Finally, the diameter of these digraphs is obtained for values of n and k which include almost all values for which they are Cayley digraphs.