  We study separability properties in a 5-dimensional set of states of quantum
systems composed of three subsystems of equal but arbitrary finite Hilbert
space dimension d. These are the states, which can be written as linear
combinations of permutation operators, or, equivalently, commute with unitaries
of the form UxUxU. We compute explicitly the following subsets and their
extreme points: (1) triseparable states, which are convex combinations of
triple tensor products, (2) biseparable states, which are separable for a
twofold partition of the system, and (3) states with positive partial transpose
with respect to such a partition. Tripartite entanglement is investigated in
terms of the relative entropy of tripartite entanglement and of the trace norm.
