It is well known that symmetry-breaking orders are described by group theory. For bosonic SPT phases with pure gauge anomalous boundary, it was shown that they are classified by group cohomology theory: those (d+1)D SPT states with symmetry G are labeled by the elements in group cohomology class H d + 1 [ G , U ( 1 ) ] . For other (d+1)D SPT states with mixed gauge-gravity anomalous boundary, they can be described by ⊕ ⊕ k = 1 d H k [ G , i T O d + 1 − − k ] , where i T O d + 1 is the Abelian group formed by (d+1)D topologically ordered phases that have no non-trivial topological excitations (referred as iTO phases).