The small quantum groups u_q(g) are finite-dimensional quotients of quantum universal enveloping algebras U_q(g) at a root of unity q for g a semisimple complex Lie algebra. After the work of Lusztig, the representation theory of these quantum objects was intensively studied because of its relation with the representation theory of semisimple algebraic groups in positive characteristic. In this talk, I will present some results on the representation theory of what we call "generalized" small quantum groups. A particular feature of these objects is that the role of the corresponding Cartan subalgebra is played by a finite non-abelian group. Nevertheless, they still admit a triangular decomposition and share similar properties with the standard quantum groups, like the existence of weights (that are no longer one-dimensional) and Verma modules. This talk is based on a joint work with Cristian Vay [Simple modules of small quantum groups at dihedral groups, Doc. Math. 29 (2024), 1–38].