  We analyse in depth an $S_3$-invariant nearest-neighbor quantum chain in the
region of a U(1)-invariant self-dual multicritical point. We find four distinct
proximate gapped phases. One has three-state Potts order, corresponding to
topological order in a parafermionic formulation. Also nearby is a phase with
"representation" symmetry-protected topological (RSPT) order. Its dual exhibits
an unusual "not-A" order, where the spins prefer to align in two of the three
directions. Within each of the four phases, we find a frustration-free point
with exact ground state(s). The exact RSPT ground state is similar to that of
Affleck-Kennedy-Lieb-Tasaki, whereas its dual states in the not-A phase are
product states, each an equal-amplitude sum over all states where one of the
three spin states on each site is absent. A field-theory analysis shows that
all transitions are in the universality class of the critical three-state Potts
model. They provide a lattice realization of a flow from a free-boson field
theory to the Potts conformal field theory.
