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The fullerene-like structures can be characterized not only by the classical topology (12 pentagonal and rest hexagonal rings), but can also contain 4-membered rings (typically forbidden due to strong steric stresses). Here we report the numbers and the symmetry point group statistics for all  C20 – C60  fulleroids with 4-membered rings (141325 in total). The most symmetrical structures with 6 to 120 automorphism group orders (251 in total) are drawn in Schlegel diagrams and characterized by facet sets and symmetry point groups in the list. Among them, 19 series of tubulen-like structures constructed from pairs of the same caps separated by belts of hexagons have been found. The obtained results are of practical interest and can be used for the description of the topologies of polyoxometallates with the fullerene-related structures.

Toward the complete set of fulleroids with four-membered rings: Predicted topologies and symmetry statistics for C n ( n = 20–60)