  We study the behavior of asymptotically free (AF) spin and gauge models when
their continuous symmetry group is replaced by different discrete non-Abelian
subgroups. Precise numerical results with relative errors down to O(0.1%)
suggest that the models with large subgroups are in the universality class of
the underlying original models. We argue that such a scenario is consistent
with the known properties of AF theories. The small statistical errors allow a
detailed investigation of the cut-off effects also. At least up to correlation
lengths ~300 they follow effectively an O(a) rather than the expected O(a^2)
form both in the O(3) and in the dodecahedron model.
