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We study the Z2 topologically ordered surface state of three-dimensionalbosonic SPT phases with the discrete symmetries G1 x G2. It has been arguedthat the topologically ordered surface state cannot be realized on a purelytwo-dimensional lattice model. We carefully examine the statement and show thatthe surface state should break G2 if the symmetry G1 is gauged. This manifeststhe conflict of the symmetry G1 and G2 on the surface of the three-dimensionalSPT phase. Given that there is no such phenomena in the purely two-dimensionalmodel, it signals that the symmetries are encoded anomalously on the surface ofthe three-dimensional SPT phases and that the surface state can never berealized on the purely two-dimensional models.

Conflicting Symmetries in Topologically Ordered Surface States of Three-dimensional Bosonic Symmetry Protected Topological Phases