  We say that a group G is a cube group if it is generated by a set S of
involutions such that the corresponding Cayley graph Cay(G,S) is isomorphic to
a cube. Equivalently, G is a cube group if it acts on a cube such that the
action is simply-transitive on the vertices and the edge stabilizers are all
nontrivial. The action on the cube extends to an orthogonal linear action,
which we call the geometric representation. We prove a combinatorial
decomposition for cube groups into products of 2-element subgroup, and show
that the geometric representation is always reducible.
