From a combinatorial point of view one can define a fullerene to be a 3-valent 3-connected graph with exactly 12 faces which are 5-gons (pentagons) and h 6-gons (hexagons). By Steinitz's Theorem fullerenes which exist as graphs can be realized by convex polyhedra. Branko Grunbaum and Theodore Motzkin showed, The number of hexagons and the simplicity of geodesics on certain polyhedra, Canadian J. Math., 15 (1963) 744-751, that the admissible values of h for such graphs are all non-negative integers h except h = 1. Other proofs of this, given by construction, show other features than what Grunbaum and Motzkin did. (For references see article listed below.)
What are the symmetry groups which can arise as the automorphism groups of fullerene graphs? There are only 28 such groups and they are listed on page 36 of the book: Geometry of Chemical Graphs: Polycylces and Two-Faced Maps, by Michel Deza, and Mathieu Dutour Sikiric, Cambridge U. Press, 2008. By a theorem of Peter Mani, these fullerene graphs can be realized by 3-dimensional polyhedra with the full automorphisms group of the graph as the group of isometries of the realizing polyhedron.
For further discussion of fullerenes and some open problems about fullerene graphs see:
Malkevitch, J., Geometrical and Combinatorial Questions about Fullerenes, in Discrete Mathematical Chemistry, (P. Hansen, P. Fowler, M. Zheng, eds.), Volume 51, DIMACS Series in Discrete Mathematics and Computer Science, AMS, Providence, 2000, pp. 261-266.
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