  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 Isomorphism 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 ${\sf GI}$-complete.
