The full symmetry and irreducible representations of nanotori

The full symmetry groups of carbon nanotori are investigated. It is shown that that the symmetry group of a chiral (n₁, n₂) nanotorus is isomorphic to D_2mq/n, where m and q are the number of lattice points on the torus circumference vector and the number of graphene hexagons in the nanotorus unit cell, respectively, and n = gcd(n₁, n₂). It is also shown that the symmetry group of zigzag and armchair (achiral) nanotori is , where D_2k and are the dihedral group of order 2k and the cyclic group of order k, respectively. The irreducible representations and characters of these groups are discussed.