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We construct two-dimensional conformal field theories with a ZN symmetry, based on the second solution of Fateev–Zamolodchikov for the parafermionic chiral algebra. Primary operators are classified according to their transformation properties under the dihedral group (ZN×Z2, where Z2 stands for the ZN charge conjugation), as singlets, ⌊(N−1)/2⌋ different doublets, and a disorder operator. In an assumed Coulomb gas scenario, the corresponding vertex operators are accommodated by the Kac table based on the weight lattice of the Lie algebra B(N−1)/2 when N is odd, and DN/2 when N is even. The unitary theories are representations of the coset SOn(N)×SO2(N)/SOn+2(N), with n=1,2,… . We suggest that physically they realize the series of multicritical points in statistical systems having a ZN symmetry. [Copyright &y& Elsevier]