The next series of symmetry groups [2+,2q] (Dqd) with q = 5,6 we obtain in the same way, from 4-valent graphs with q-gonal 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 C20 with the icosahedral symmetry group G and G' = D5d, that can be used as a building block for the complete class of nanotubes C40, C60, C80,... with G = D5d, that can be obtained from C20 by "gluing" the pentagonal bases. In the same way, fullerene C24 obtained for q = 6 and k = 0 can be used as the building block for the nanotubes C48, C72, C96,... The geometrical structure of the nanotube class with G = Dqd (q = 5,6) permits the edge coloring that preserves symmetry, so there always exist the isomers with G = G¢.
If the 3-rotation axis contains the opposite vertices of a fullerene, we have biconical fullerenes (e.g., C26, C56) with G = D3h, G = D3d, 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 C26, G = D3h, G' = C2v.
Proceeding in the same way, it is possible to find or construct fullerene representatives of other symmetry groups from the category G320: biconical C32 with G = D3, biconical C38 or conical C34 with G = C3v, conical C46 with G = C3 (Boo, 1992), or the infinite class of cylindrical fullerenes C42, C48, C54,... with G = D3. In general, edge coloring of 3-valent 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.