Classification of finite groups that admit an oriented regular representation

This is the third, and last, of a series of papers dealing with oriented regular representations. Here we complete the classification of finite groups that admit an oriented regular representation (or ORR for short), and give a complete answer to a 1980 question of Laszlo Babai: "Which [finite] groups admit an oriented graph as a DRR?" It is easy to see and well-understood that generalised dihedral groups do not admit ORRs. We prove that, with 11 small exceptions (having orders ranging from 8 to 64), every finite group that is not generalised dihedral has an ORR.