 Holonomy groups of closed orientable flat three-manifolds

These are precisely the faithful linear holonomy groups of closed orientable Euclidean three-manifolds. Cyclic cases arise from planar lattice rotations combined with axial screw translations. The noncyclic case is generated by two perpendicular half-turn screws. The list refers to the maximal translation lattice, not to arbitrary finite quotients of a fundamental group.