Symmetry Elements of a 3D-Object

The simplest point group C₁ contains only one trivial symmetry operation, the identity operation E. This is the point group of all asymmetrical molecules and molecular conformations. These molecules must be chiral:

C1	C1	C1

If a molecule possesses only an inversion center i and no other symmetry elements, it belongs to the point group C_i. Examples of this group are commonly found with meso-type structures such as meso-tartaric acid.

Ci	Ci

This point groups has a mirror plane σ as the only symmetry element. Many (flat) aromatic molecules belong to this group.

Cs	Cs	Cs	Cs

Molecules in this group have a single n-fold rotation axis C_n as their symmetry element. These compounds must be chiral, and many of the most important ligands used in stereoselective synthesis of organic chemistry belong to the C₂ point group.

C2	C2	C2	C2	C3	C3	C3	C4	C6

Molecules having a n-fold rotation axis C_n and n vertical mirror planes σ_v belong to the C_nv point group. Linear molecules which do not possess an inversion center or a horizontal mirror plane σ_h belong to the C_∞v point group, because all rotations about their axis are symmetry operations (conical molecules).

C2v	C2v	C2v	C2v	C3v	C3v	C4v	C∞v

Molecules belonging to the C_nh point group feature a n-fold rotation axis C_n and a horizontal mirror plane σ_h.

C2h	C2h	C2h	C3h

A molecule that has a n-fold principal axis and n twofold axes perpendicular to the principal axis belongs to the D_n point group. All molecules in this point group must be chiral.

D2	D3	D3	D3

If, in addition to a n-fold principal axis and n twofold axes perpendicular to it, a horizontal mirror plane σ_h is present in a molecular structure, the point group is described as D_nh. The D_∞h point group includes all linear molecules with an center of inversion, which also implies a horizontal mirror plane σ_h (cylindrical molecules).

D2h	D2h	D2h	D3h	D4h	D5h	D6h	D6h	D∞h

The D_nd point groups include the symmetry elements of D_n and n dihedral mirror planes σ_d bisecting the angles formed by pairs of C₂ axes.

D2d	D2d	D2d	D3d	D3d	D5d

Molecules which have not been classified by one of the above C or D point groups, but which possess one rotary-reflection axis S_n only (n must be even and ≥ 4), belong to the S_n point groups. Molecules of point groups S_n with n > 4 are rare.

S4	S4	S4	S4	S6

The tetrahedral point groups are characterized by the presence of four C₃ principal axes (and three C₂ axes). The T_d group is the point group of a regular tetrahedron. If, in addition, a center of inversion is present, the point group is T_h (molecules of this group do not look like tetrahedrons, but retain the rotational symmetry of a tetrahedron). All objects possessing the rotational symmetry of a tetrahedron, but no plane of reflection or center of inversion are based on the simpler point group T. Molecules of this group must be chiral, and examples are very rare.

T	Th	Td	Td	Td	Td

The octahedral point groups O and O_h feature three C₄ principal axes (and four C₃ axes as well as multiple C₂ axes). A regular octahedron and a cube both belong to the O_h point group. In analogy to the tetrahedral point group T, the octahedral group O retains the rotational symmetry of an regular octahedron, but none of its planes of reflection or the center of inversion. Examples of molecules belonging to the O point group are extremely rare (these molecules must be chiral!).

Oh	Oh	Oh	Oh

The icosahedral point groups I and I_h posses six C₅ principal axes (amongst 10 C₃ and 15 C₂ axes). In analogy to the octahedral point groups O and O_h the I group does contain all rotational symmetry elements, but no mirror planes or center of inversion. Molecules belonging to this chiral point groups are extremely rare. Shapes belonging to the more symmetrical I_h point groups are the dodecahedron and the icosahedron.

Ih	Ih	Ih

The K_h point group resembles the symmetry of perfect spheres. Obviously, only atoms but no molecules belong to this point group.