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In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O(d).
In the classification of crystals, to each space group is associated a crystallographic point group by "forgetting" the translational components of the symmetry operations. That is, by turning screw rotations into rotations, glide reflections into reflections and moving all symmetry elements into the origin.
The point group symmetry of a molecule is defined by the presence or absence of 5 types of symmetry element. Symmetry axis: an axis around which a rotation by results in a molecule indistinguishable from the original. This is also called an n-fold rotational axis and abbreviated C n.
The elements of this symmetry group should not be confused with the "symmetry element" itself. Loosely, a symmetry element is the geometric set of fixed points of a symmetry operation. For example, for rotation about an axis, the points on the axis do not move and in a reflection the points that remain unchanged make up a plane of symmetry.
D nh is the symmetry group for a "regular" n-gonal prism and also for a "regular" n-gonal bipyramid. D nd is the symmetry group for a "regular" n-gonal antiprism, and also for a "regular" n-gonal trapezohedron. D n is the symmetry group of a partially rotated ("twisted") prism. The groups D 2 and D 2h are noteworthy in that there is no special ...
The symmetry elements are ordered the same way as in the symbol of corresponding point group (the group that is obtained if one removes all translational components from the space group). The symbols for symmetry elements are more diverse, because in addition to rotations axes and mirror planes, space group may contain more complex 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.
For each non-linear group, the tables give the most standard notation of the finite group isomorphic to the point group, followed by the order of the group (number of invariant symmetry operations). The finite group notation used is: Z n: cyclic group of order n, D n: dihedral group isomorphic to the symmetry group of an n–sided regular ...