Examples of solvable and nilpotent finite quantum groups

We prove the solvability and nilpotency of Kac–Paljutkin’s finite quantum group and Sekine quantum groups and we classify the solvable series of Kac–Paljutkin’s finite quantum group via Cohen–Westreich’s Burnside theorem. Some semisimple quasitriangular Hopf algebras of dimensions $2pq$ are also studied. In Appendix A, we give a direct computation of the universal $R$-matrices for Kac–Paljutkin’s $8$-dimensional finite quantum group.