Classification of molecules
according to symmetry
If analysis of a molecule yielded a list of all present symmetry elements, such a list is the basis to form classes of molecules with common symmetry. Such a class is called a symmetry point group. Any molecule is unambiguously a member of one single point group. Though the point groups relevant in chemistry are numerous, we are able to work straightforward within this field as these groups are characterized by sets of increasingly complicated symmetry operations.
Two notations are in use, the Schönflies system (according to Arthur Moritz Schönflies, 17.4.1853  27.5.1928). and the HermannMauguin system. The former is more common in the discussion of individual molecules, the latter is used almost exclusively in the discussion of crystal symmetry. Note that both are expressions of one single system and they only differ with respect to semantics.
List of point groups

The groups C_{1}, C_{i} and C_{s}. If a molecule (or any object) possesses no symmetry element other than the identity, it belongs to C_{i}. Only a rotation about 360° produces an identical image of the molecule. As this operation establishes the initial orientation, any body, asymmetric as it may be, meets this criterion. If a molecules possesses the identity and the inversion as the only elements, it belongs to the group C_{i}. If it possesses a plane of symmetry as the only element apart from the identity, it is classified as belonging to the group C_{s}.

The groups C_{n}. If a molecule possesses the identity element and an nfold axis of symmetry, it belongs to the group C_{n}. (Note that C_{n} is now playing a triple role: it is a label for one of the symmetry elements present and denotes the corresponding operation as well as the name of the group.)

The groups C_{nv}. Objects in these groups possess a C_{n} axis and n vertical reflection planes σ_{v}.

The groups C_{nh}. Objects possessing a C_{n}axis and a perpendicular horizontal mirror plane belong to the group C_{nh}. Note that this is an example for a group where automatically other symmetry operations can be applied. In this case the inversion i is found too as C_{2} and σ_{h} are present.

The groups D_{n} with molecules possessing an nfold principal axis and n twofold axis perpendicular to C_{n}.

The groups D_{nh}. Molecules belong to this group if they belong to D_{n} and possess a horizontal mirror plane. All homonuclear diatomics belong to group D_{∞n} and all heteronuclear molecules to C_{∞v}.

The groups D_{nd}. The classification of D_{n} is also based on D_{n} but requires the additional presence of vertical mirror planes bisecting the angles between all the neighbouring axis C_{2}.

The groups S_{n}. Molecules with an rotatory reflection axis S_{n} belong to this group. There are only a few molecules with n < 4. Group S_{2} is identical with C_{i}.

The cubic groups T and O. A number of very important molecules possess more than one principal axis of symmetry. For example, CH_{4} possesses four C_{3} axes, one along each bond. The groups to which these belong are called the tetrahedral groups T, T_{d} and T_{h} and the octahedral groups O_{h} and O. The regular tetrahedron is a representative of T_{d}; the regular octahedron of O_{h}. If the molecule possesses the rotational symmetry of the tetrahedron or the octahedron but none of its planes of reflection, it belongs to the simpler groups T and O, respectively. The group T_{h} is slightly peculiar because it is based on T but also includes a centre of inversion.

The full rotation group R_{3} is the group of operations shown by a spherical object. An atom belongs to R_{3}, but no molecule does. Exploring the consequences of R_{3} symmetry is a very important way of applying the grouptheoretical arguments to atoms.
A flow chart flow chart provides valuable help in the classification of molecules.
Examples:

Water
The water molecule possesses one twofold rotational axis C_{2}
and two mirror planes σ_{v} and σ'_{v} parallel to this axis. Therefore, it belongs to symmetry point group C_{2v}. 

transButadiene
This molecule possesses a centre of inversion i, one twofold rotational axis C_{2} and a mirror plane σ_{h} which is horizontal, i.e. perpendicular to the axis. Transbutadiene falls in symmetry point group C_{2h}. 



Benzene
The features of this molecule are a centre of inversion i, rotational axes C_{2} and C_{2}' and C_{6}, a horizontal mirror plane σ_{h} perpendicular to C_{6} and mirror planes σ_{v} and σ_{d} parallel to C_{6}. This leads to a classification in point group D_{6h}. 
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