  We provide a classification of compact quantum groups, which can be obtained
by the Woronowicz construction, when the arrays used in the twisted determinant
condition are extensions of functions on permutations. General properties of
such quantum groups are revealed with the aid of operators intertwining tensor
powers of the fundamental corepresentation. Two new families of quantum groups
appear: three versions of the quantum group $SU_q(3)$ with complex $q$ and a
non-commutative analogue of the semi-direct product of two-dimensional torus
with the alternating group $A_3$.
