  The spin-1/2 Hamiltonian for two coupled isosceles Heisenberg triangles,
which is well suited for describing the V$_6$-type magnetic molecules, is
studied by exact diagonalization. The quantum phase transition diagram, at zero
temperature, is obtained as a function of the theoretical parameters. The zero
temperature magnetization is also obtained as a function of the external
magnetic field. The thermodynamic behavior of the magnetization, entropy,
susceptibility, and specific heat, as a function of temperature, are also
computed and the corresponding magnetocaloric effect analyzed for various
values of the Hamiltonian parameters.
