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We investigate the phase diagram of a quantum spin-1 chain whose Hamiltonianis invariant under a global onsite $A_4$, translation and lattice inversionsymmetries. We detect different gapped phases characterized by SPT order andsymmetry breaking using matrix product state order parameters. We observe arich variety of phases of matter characterized by a combination of symmetrybreaking and symmetry fractionalization and also the interplay between theonsite and spatial symmetries. Examples of continuous phase transitionsdirectly between topologically nontrivial SPT phases are also observed.

Detection of gapped phases of a 1D spin chain with onsite and spatial symmetries