Hermann-Mauguin/International Notations for Symmetry Elements
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In crystallography and geometry, Hermann–Mauguin notation has been used to describe the symmetry elements in point groups, plane groups and space groups. It was named after the German crystallographer Carl Hermann because he introduced it in 1928 and the French mineralogist Charles-Victor Mauguin because he modified it in 1931. The Hermann–Mauguin notation, compared with the Schoenflies notation, is preferred in crystallography because it includes translational symmetry elements (glide planes and screw-axis) and specifies the directions of the symmetry axes. This notation is often called international notation because it has been used as standard by the International Tables For Crystallography since the first edition in 1935.

Some conventional rules for describing the symmetry elements have been applied in Hermann–Mauguin notation:
i) If a rotation axis n and a mirror plane m are in the same direction (i.e. the mirror plane is perpendicular to axis n), then they are denoted as fraction or n/m.
ii) If two or more axes have the same direction, only the axis with higher symmetry is listed in the notation. Note that the higher symmetry is the symmetry with the axis that generates a pattern with more points. Rotation axes 3, 4, 5, 6, 7, 8 generate 3-, 4-, 5-, 6-, 7-, 8-point patterns, respectively. For instance, 3/m () is equivalent to -6. Because -6 generates 6 points and 3 generates only 3; therefore, -6 should be selected. Another case is when both 3 and -3 axes are present, -3 should be selected.
iii) If both rotation and roto-inversion axes have the same direction and have equal symmetry, the rotation axis should be selected. For instance, when both 4/m and -4/m axes are present, 4/m should be selected because both 4 and -4 generate four points. Another example is if 2, 3, 6, -3, and -6 axes are present, then 6 axis should be selected because it is a rotation axis and has higher symmetry (axes -3, -6, and 6 all generate 6-point patterns).
iv) If the symbol contains three positions, then they denote the symmetry elements in X, Y, Z directions, respectively.

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