The next series of symmetry groups [2+,2q] (D_{qd}) with q = 5,6 we obtain in the same way, from 4valent graphs with qgonal bases, 2q triangular and 2kq quadrilateral faces (k = 1,2,... for q = 5; k = 0,1,2,... for q = 6). As the limiting case, for q = 5 and k = 0, we obtain C_{20} with the icosahedral symmetry group G and G' = D_{5d}, that can be used as a building block for the complete class of nanotubes C_{40}, C_{60}, C_{80},... with G = D_{5d}, that can be obtained from C_{20} by "gluing" the pentagonal bases. In the same way, fullerene C_{24} obtained for q = 6 and k = 0 can be used as the building block for the nanotubes C_{48}, C_{72}, C_{96},... The geometrical structure of the nanotube class with G = D_{qd} (q = 5,6) permits the edge coloring that preserves symmetry, so there always exist the isomers with G = G¢. If the 3rotation axis contains the opposite vertices of a fullerene, we have biconical fullerenes (e.g., C_{26}, C_{56}) with G = D_{3h}, G = D_{3d}, respectively. After the edge coloring, symmetry must be disturbed, and for the biconical fullerenes G' is always a proper subgroup of G. For example, for C_{26}, G = D_{3h}, G' = C_{2v}. Proceeding in the same way, it is possible to find or construct fullerene representatives of other symmetry groups from the category G_{320}: biconical C_{32} with G = D_{3}, biconical C_{38} or conical C_{34} with G = C_{3v}, conical C_{46} with G = C_{3} (Boo, 1992), or the infinite class of cylindrical fullerenes C_{42}, C_{48}, C_{54},... with G = D_{3}. In general, edge coloring of 3valent graphs changes symmetry of all conical or biconical fullerenes mentioned, so their geometrical symmetry is always higher then the chemical. All nanotubes, conical and biconical fullerenes described in this subsection can be analyzed by converting them to corresponding alternating KLs, where the geometrical symmetry properties can be connected with the corresponding symmetries of KLs.
