Categorisation by Class: Non-Crystallographic

Cubic (T T_h O T_d O_h)

Tetragonal (C₄ S₄ C_4h D₄ C_4v D_2d D_4h)

Orthorhombic (D₂ C_2v D_2h)

Monoclinic (C₂ C_s C_2h)

Triclinic (C₁ C_i)

Trigonal (C₃ S₆ D₃ C_3v D_3d)

Hexagonal (C₆ C_3h C_6h D₆ C_6v D_3h D_6h)

Non-crystallographic *

Operations Order Schönflies
Symbol International
Symbol Full
Symmetry
Symbol Correlation
Table Irred. Rep.
products × i Isomorph.
with

E, 2 C infinity ∞ C_∞ ∞ - - C_∞h

E, 2, i, 2 S infinity ∞ C_∞h ∞/m - - C_∞h

E, 2 C infinity ,∞σ_v ∞ C_∞v ∞m - - D_∞h

E, 2,∞σ_v, i, 2 S infinity ,∞C₂ ∞ D_∞h ∞/mm - - D_∞h

E, C₅, C₅², C₅³, C₅⁴ 5 C₅ 5 - ⊗ S₁₀

E, S₈, C₄, S₈³, C₂, S₈⁵, C₄³, S₈⁷ 8 S₈ - - ⊗ C_8h

E, 2C₅, 2C₅², 5C₂ 10 D₅ - - ⊗ D_5d C_5v

E, 2C₅, 2C₅², 5σ_v 10 C_5v - - ⊗ D_5d D₅

E, C₅, C₅², C₅³, C₅⁴,σ_h, S₅, S₅⁷, S₅³, S₅⁹ 10 C_5h - - ⊗ C_10h

E, 2S₈, 2C₄, 2S₈³, C₂, 4C₂’, 4σ_d 16 D_4d - - ⊗ D_8h

E, 2C₅, 2C₅², 5C₂, i, 2S₁₀, 2S₁₀³, 5σ_d 20 D_5d - - ⊗ D_5d D_5h

E, 2C₅, 2C₅², 5C₂,σ_h, 2S₅, 2S₅², 5σ_d 20 D_5h - - ⊗ D_10h D_5d

E, 2S₁₂, 2C₆, 2S₄, 2C₃, 2S₁₂⁵, C₂, 6C₂’, 6σ_d 24 D_6d - - ⊗ D_12h

E, 12C₅, 12C₅², 20C₃, 15C₂ 60 I - - ⊗ I_h

E, 12C₅, 12C₅², 20C₃, 15C₂, i, 12S₁₀, 12S₁₀³, 20S₆, 15σ 120 I_h - - ⊗ I_h

All

This table lists point group symmetries along with their symmetry operations, the order of the group (i.e. the number of symmetry operations) and common notations. maps to symbol links to a correlation table, and ⊗ links to tables of products of irreducible representations. The group produced by combination with inversion is listed under "× i". This, in the case of crystolographic point groups, is the Laue class which corresponds to the symmetry of reciprocal space. Isomorphic groups are also listed where character tables are available.

Revised: