Cluster Measure Symmetry
This report describes, for the selected structure, the point group of symmetry, the symmetry operations and the symmetry elements. When Periodicity is set to Yes and various crystallographic point groups are possible, they are listed in sequence of increasing symmetry.

### Point Group

The chemical or crystallographic point group of symmetry is reported here. When Periodicity is set to No (the default), the chemical group is shown, using the Schoenflies notation (or Undefined, if no point group can be determined).

Chemical groups for linear systems, with a rotation axis of infinite order, are named C0v and D0h. The spherical group, corresponding to a single atom, is named Kh.

When Periodicity is set to Yes, the group is named using first the International and then the Schoenflies notation (or Undefined, if no point group can be determined. In this case, rotation axes of order different from 6, 4, 3, 2 are discarded, all the other elements are shown).

Gamgi can find axes with any rotation order, so any chemical (infinite) or crystallographic (32) group of symmetry can be determined.

When users require the crystallographic point group, Gamgi determines first the chemical group and then applies the periodic restrictions to obtain the point group in a crystal with the highest possible symmetry. When more than one option is available, Gamgi shows the various solutions. For example, a C24 rotation axis in a molecule can be restricted to 6 or 4 axes in a periodic crystal.

Gamgi tries to find all the symmetry elements independently, and in general each of these elements requires a different tolerance. Thus, for a given tolerance, some elements may be recognized and others may go missing, resulting in an inconsistent set of symmetry elements. When this happens, the group is reported Undefined. The solution is to increase the tolerance (so valid elements might be found) or to decrease it (so fake elements might be discarded).

### Symmetry Operations

The complete set of symmetry operations, generated from the symmetry elements found (forming a group, in the mathematical sense), is reported in abreviated format. For example, a C4 axis generates operations C41, C42 (equal to C21), C43 and C44 (equal to E), so only two C4 operations are new, described as 2C4.

In groups with infinite rotation orders, C0v and D0h, rotation operations are presented, as 2C0 and 2S0, symbolizing the two directions of rotation. In these cases, a single mirror plane m is considered (although they are infinite). For the Kh spherical group, a single rotation axis and plane are considered (although they are infinite).

### Symmetry Elements

All symmetry elements that were found are individually reported here. The inversion center is described by its coordinates. Mirror planes are described indicating the corresponding normal vectors. For rotation axes, normal and improper, the rotation order is reported, plus the normal vector describing the axis, starting from the center.

For all symmetry elements, the error produced when applying the operations of symmetry, is reported (this error is smaller than the tolerance, otherwise the element would have been rejected).

Rotation axes are sorted according to increasing rotation order, so more symmetric axes come first. When present, infinite order axes are always the first in the list. Mirror planes and axes with the same rotation order are sorted according to decreasing element error, so better defined elements come first.

When present, a horizontal mirror plane is always listed before the other planes. When present, a C2 axis along the main direction is always listed before the other C2 axes.
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