For denoting different categories of symmetry groups, we will use Bohm symbols [16]. In a symbol G_nst…, the first subscript n represents the maximal dimension of space in which the transformations of the symmetry group act, while the following subscripts st… represent the maximal dimensions of subspaces remaining invariant under the action of transformations of the symmetry group, that are properly included in each other. With regard to their symmetry, general fullerenes belong to the category of point groups G₃₀. The category G₃₀ consists of seven polyhedral symmetry groups without invariant planes or lines: [3,3] or T_d, [3,3]⁺ or T, [3,4] or O_h, [3,4]⁺ or O, [3⁺,4] or T_h, [3,5] or I_h, [3,5]⁺ or I, and from seven infinite classes of point symmetry groups with the invariant plane (and the line perpendicular to it in the invariant point): [q] or C_qv, [q]⁺ or C_q, [2⁺,2q⁺] or S_2q, [2,q⁺] or C_qh, [2,q]⁺ or D_q, [2⁺,2q] or D_qd, [2,q] or D_qh, belonging to the subcategory G₃₂₀ [5]. For the groups of the subcategory G₃₂₀, in the case of rotations of order q>2, the invariant line (i.e. the rotation axis) may contain 0,1 or 2 vertices of a general fullerene. According to this, among all general fullerenes with a geometrical symmetry group G belonging to G₃₂₀, from the topological point of view we could distinguish, respectively, cylindrical fullerenes (nanotubes), conical and biconical ones. 

We could simply conclude that for polyhedral 5/6 fullerenes G could be only [3,3] (T_d ), [3,3]⁺ (T), [3,5] (I_h), [3,5]⁺ (I), because of their topological structure (n₅=12), incompatible with the octahedral symmetry group [3,4] (O_h) or its polyhedral subgroups. In the case of nanotubes (or cylindrical fullerenes) we have infinite classes of 5/6 fullerenes with the geometrical symmetry group [2,q] (D_qh) and [2⁺,2q] (D_qd), and the same chemical symmetry. The first infinite first class of cylindrical nanotubes with G=G'=D_5h we obtain from a cylindrical 3/4/5 4-regular graph with two pentagonal bases, 10 triangular and 5(2k+1) quadrilateral faces (k=0,1,2,…) and with the same symmetry group (Fig. 11). By the vertex bifurcation preserving its symmetry, we obtain the infinite class of nanotubes C₃₀, C₅₀, C₇₀,…, with C₇₀ as the first of them satisfying IPR. Certainly, the geometrical structure of C₇₀ admits different edge colorings (i.e. chemical isomers). Starting from any two of them (Fig. 12) by "collapsing" (the inverse of "vertex bifurcation", i.e. by deleting digons) we could obtain different generating 4-regular graphs. This example of two different C₇₀ isomers, with the same geometrical structure, and with the same G and G', shows that for the exact recognition of fullerene isomers we need to know more than their geometrical and chemical symmetry (see Part 3). In the same way, from 4-regular graphs with two hexagonal bases, 12 triangular and 6(2k+1) quadrilateral faces (k=0,1,2,…) we obtain the infinite class of fullerenes C₃₆, C₆₀ C₈₄,… with the symmetry group G=G'=D_6h (Fig. 13). 
  

 

 

 

The next symmetry groups [2⁺,2q] (D_qd) with q=5,6 we obtain in the same way, from 4-regular graphs with q-gonal bases, 2q triangular and 2kq quadrilateral faces (k=1,2,… for q=5; k=0,1,2,… for q=6) (Fig. 14). As the limiting case, for q=5 and k=0, we obtain C₂₀ with the icosahedral symmetry group G, but with G'=D_5d, that could be used as the "brick" for the complete class of nanotubes C₄₀, C₆₀,C₈₀,… with G= D_5d, where all of them could be obtained from C₂₀ by "gluing" the pentagonal bases (Fig. 15). In the same way, fullerene C₂₄ obtained for q=6 and k=0 could be used as the building block for the nanotubes C₄₈, C₇₂,C₉₆,… The geometrical structure of nanotube class with G=D_qd (q=5,6) permits the edge coloring preserving the symmetry, so there always exist their 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=D_3h, G=D_3d, respectively (Fig. 16). Certainly, after the edge coloring, their symmetry must be disturbed, and for them G' is always a proper subgroup of G. For example, for C₂₆ (Fig. 16), 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 G₃₂₀: biconical C₃₂ with G=D₃, biconical C₃₈ or conical C₃₄ with G=C_3v, conical C₄₆ with G=C₃ [17], or the infinite class of cylindrical fullerenes C₄₂, C₄₈, C₅₄… with G=D₃ (Fig. 17). In general, after edge coloring of their 3-regular graphs, symmetry could not be preserved in all conical or biconical fullerenes mentioned, so their geometrical symmetry is always higher than the chemical.