  We construct and classify all groups, given by triangular presentations
associated to the smallest thick generalized quadrangle, that act simply
transitively on the vertices of hyperbolic triangular buildings of the smallest
non-trivial thickness. Our classification shows 23 non-isomorphic torsion free
groups (obtained in an earlier work) and 168 non-isomorphic torsion groups
acting on one of two possible buildings with the smallest thick generalized
quadrangle as the link of each vertex. In analogy with the Euclidean case, we
find both torsion and torsion free groups acting on the same building.
