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The S 2 group is the same as the C i group in the nonaxial groups section. S n groups with an odd value of n are identical to C nh groups of same n and are therefore not considered here (in particular, S 1 is identical to C s). The S 8 table reflects the 2007 discovery of errors in older references. [4] Specifically, (R x, R y) transform not as ...
Each crystallographic point group defines the (geometric) crystal class of the crystal. The point group of a crystal determines, among other things, the directional variation of physical properties that arise from its structure, including optical properties such as birefringency , or electro-optical features such as the Pockels effect .
The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a concise form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on any representative of the respective conjugacy class of G (because characters are class functions).
The table below organizes the space groups of the monoclinic crystal system by crystal class. It lists the International Tables for Crystallography space group numbers, [ 2 ] followed by the crystal class name, its point group in Schoenflies notation , Hermann–Mauguin (international) notation , orbifold notation, and Coxeter notation, type ...
A further sub-division into systems is defined by the rotational group G in the leftmost column then into rows of Laue classes. Every point group in a Laue class has exactly the same abstract group structure except the centred group in the rightmost column which is the direct product of the rotational group with inversion.
Each point group can be represented as sets of orthogonal matrices M that transform point x into point y according to y = Mx. Each element of a point group is either a rotation (determinant of M = 1), or it is a reflection or improper rotation (determinant of M = −1). The geometric symmetries of crystals are described by space groups, which ...
For example, 4 1 /a means that the crystallographic axis in question contains both a 4 1 screw axis as well as a glide plane along a. In Schoenflies notation, the symbol of a space group is represented by the symbol of corresponding point group with additional superscript. The superscript doesn't give any additional information about symmetry ...
Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry. This article lists the groups by Schoenflies notation, Coxeter notation, [1] orbifold notation, [2] and order.