  Hypercubic groups in any dimension are defined and their conjugate
classifications and representation theories are derived. Double group and
spinor representation are introduced. A detailed calculation is carried out on
the structures of four-dimensional cubic group $O_4$ and its double group, as
well as all inequivalent single-valued representations and spinor
representations of $O_4$. All representations are derived adopting Clifford
theory of decomposition of induced representations. Based on these results,
single-valued and spinor representations of the orientation-preserved subgroup
of $O_4$ are calculated.
