Automorphisms of the Cube n

Consider a hypergraph H n where the vertices are points of the d-dimensional combinatorial cube n 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 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 n there exists an automorphism of H n preserving the colors. We show that this problem is GI-complete.