Finite quantum groups and quantum permutation groups

We give examples of finite quantum permutation groups which arise from the twisting construction or as bicrossed products associated to exact factorizations in finite groups. We also give examples of finite quantum groups which are not quantum permutation groups: one such example occurs as a split abelian extension associated to the exact factorization S4=Z4S3 and has dimension 24. We show that, in fact, this is the smallest possible dimension that a non-quantum permutation group can have.