  Polyhedral nematics are examples of exotic orientational phases that possess
a complex internal symmetry, representing highly non-trivial ways of rotational
symmetry breaking, and are subject to current experimental pursuits in
colloidal and molecular systems. The classification of these phases has been
known for a long time, however, their transitions to the disordered isotropic
liquid phase remain largely unexplored, except for a few symmetries. In this
work, we utilize a recently introduced non-Abelian gauge theory to explore the
nature of the underlying nematic-isotropic transition for all three-dimensional
polyhedral nematics. The gauge theory can readily be applied to nematic phases
with an arbitrary point-group symmetry, including those where traditional
Landau methods and the associated lattice models may become too involved to
implement owing to a prohibitive order-parameter tensor of high rank or (the
absence of) mirror symmetries. By means of exhaustive Monte Carlo simulations,
we find that the nematic-isotropic transition is generically first-order for
all polyhedral symmetries. Moreover, we show that this universal result is
fully consistent with our expectation from a renormalization group approach, as
well as with other lattice models for symmetries already studied in the
literature. We argue that extreme fine tuning is required to promote those
transitions to second order ones. We also comment on the nature of phase
transitions breaking the $O(3)$ symmetry in general cases.
