  The phase diagram of a coupled tetrahedral Heisenberg model is obtained. The
quantum chain has a local gauge symmetry and its eigenspectrum is obtained by
the composition of the eigenspectra of spin-1/2 XXZ chains with arbitrary
distribution of spin-3/2 impurities. The phase diagram is quite rich with an
infinite number of phases with ferromagnetic, antiferromagnetic or
ferrimagnetic order. In some cases the ground state and the low lying
eigenlevels of the model can be exactly calculated since they coincide with the
eigenlevels of the exactly integrable XXZ chain. The thermodynamical properties
of the model at low temperatures is also studied through finite-size analysis.
