  We generalize the concept of cubic group into any dimension and derive their
conjugate classifications and representation theorys. Double group and spinor
representation are defined. 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.
