Abstract
Consider a hypergraph $H_n^d$ 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 $H_n^d$, 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 Automorphism problem of deciding whether for two colorings of the vertices of $H_n^d$ there exists an automorphism of $H_n^d$ preserving the colors.
We show that this problem is $\GI$-complete.