Single wall carbon nanotubes have line group symmetry.
Simetries on honeycomb Chiral tube: 5th line group family

Depending on the single wall constituents and their relative position, multiwall tubes may be both periodic (with line group symmetry) or aperiodic (point group symmetry). The high symmetry of nanotubes tremendously simplified studies of physical tensors, selection rules, potentials, electronic and vibrational bands... Optical activity of the chiral carbon tubes only is predicted together with geometry of optical axes. Conserved quantum numbers are translational, chiral and residual angular momenta, together with parities. Invariant potentials are used to analyze stability of multiwall tubes and the chiral curents. Modified group projector technique gives symmetry assignation and topology of the energy bands. Different tubes have highly incompatible symmetries. This fact is used to predict extremely low friction between the walls in the multiwall tubes, as a result of specific symmetry breaking: even free sliding of the incommensurate walls occurs as a Goldstone mode.

Zig-zag tub5: 13th line group family Armchair tube: 13th line group family

The PC program POLSym® efficiently performs all these calculations.