We study algebraic classification and rigidity properties of ABC group actions on the three torus \(T^3\), by linear and affine transformations. The linear part of such an action is an ABC subgroup of SL(3, Z). We investigate when such a linear ABC action on \(T^3\)  can be extended to an affine action that has no identity factors. For a particular class of such actions, we show KAM rigidity; the main reason for the existence of the conjugacy is KAM rigidity of the parabolic \(Z^2\) action inside the ABC group action.