Is our vacuum global in a 331 model with three triplets?

We consider a 331 model, based on $\beta=-1/\sqrt{3}$, with three $SU(3)$ triplets with a softly broken $\mathbb{Z}_2$ symmetry. The resulting scalar potential is commonly used in phenomenology. We systematically determine all the potential minima and obtain the conditions under which the electroweak vacuum is global with the help of orbit space methods. For the case the electroweak vacuum is not global, we calculate bounds on the scalar couplings from metastability. We find a parametrisation of the potential couplings in terms of physical quantities and use it to show the available parameter space.