Automorphisms of the Cube n^d

Consider a hypergraph H d n where the vertices are points of the d-dimensional combinatorial cube n d and the edges are all sets of n points such that they are in one line. We study the structure of the group of automorphisms of H d n , i.e., permutations of points of n d preserving the edges. In this paper we provide a complete characterization. Moreover, we consider the Colored Cube Isomorphism problem of deciding whether for two colorings of the vertices of H d n there exists an automorphism of H d n preserving the colors. We show that this problem is GI-complete.