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.