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