  We consider Abelian-by-cyclic groups for which the cyclic factor acts by
hyperbolic automorphisms on the Abelian subgroup. We show that if such a group
acts faithfully by $C^1$ diffeomorphisms of the closed interval with no global
fixed point at the interior, then the action is topologically conjugated to
that of an affine group. Moreover, in case of non-Abelian image, we show a
rigidity result concerning the multipliers of the homotheties, despite the fact
that the conjugacy is not necessarily smooth. Some consequences for
non-solvable groups are proposed. In particular, we give new proofs/examples
yielding the existence of finitely-generated, locally-indicable groups with no
faithful action by $C^1$ diffeomorphisms of the interval.
