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In crystallography, a crystallographic point group is a three dimensional point group whose symmetry operations are compatible with a three dimensional crystallographic lattice. According to the crystallographic restriction it may only contain one-, two-, three-, four- and sixfold rotations or rotoinversions. This reduces the number of ...
In mathematics, a group is a set with a binary operation that ... For example, with the point group C 3, ... Point group C2h has the operations and transformation ...
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 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 ...
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).
For point groups that reverse orientation, the situation is more complicated, as there are two pin groups, so there are two possible binary groups corresponding to a given point group. Note that this is a covering of groups, not a covering of spaces – the sphere is simply connected, and thus has no covering spaces. There is thus no notion of ...
All of the discrete point symmetries are subgroups of certain continuous symmetries. They can be classified as products of orthogonal groups O( n ) or special orthogonal groups SO( n ). O(1) is a single orthogonal reflection, dihedral symmetry order 2, Dih 1 .
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 ...