Hermann–Mauguin notation

In geometry, Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann and the French mineralogist Charles-Victor Mauguin. This notation is sometimes called international notation, because it was adopted as standard by the International Tables For Crystallography since their first edition in 1935.
The Hermann–Mauguin notation, compared with the Schoenflies notation, is preferred in crystallography because it can easily be used to include translational symmetry elements, and it specifies the directions of the symmetry axes.

Point groups

Rotation axes are denoted by a number n – 1, 2, 3, 4, 5, 6, 7, 8, .... For improper rotations, Hermann–Mauguin symbols show rotoinversion axes, unlike Schoenflies and Shubnikov notations, that shows rotation-reflection axes. The rotoinversion axes are represented by the corresponding number with a macron, –,,,,,,,, ... . is equivalent to a mirror plane and usually notated as m. The direction of the mirror plane is defined as the direction perpendicular to it.
Hermann–Mauguin symbols show non-equivalent axes and planes in a symmetrical fashion. The direction of a symmetry element corresponds to its position in the Hermann–Mauguin symbol. If a rotation axis n and a mirror plane m have the same direction, then they are denoted as a fraction or n/m.
If two or more axes have the same direction, the axis with higher symmetry is shown. Higher symmetry means that the axis generates a pattern with more points. For example, rotation axes 3, 4, 5, 6, 7, 8 generate 3-, 4-, 5-, 6-, 7-, 8-point patterns, respectively. Improper rotation axes,,,,, generate 6-, 4-, 10-, 6-, 14-, 8-point patterns, respectively. If a rotation and a rotoinversion axis generate the same number of points, the rotation axis should be chosen. For example, the combination is equivalent to. Since generates 6 points, and 3 generates only 3, should be written instead of . Analogously, in the case when both 3 and axes are present, should be written. However we write, not, because both 4 and generate four points. In the case of the combination, where 2, 3, 6,, and axes are present, axes,, and 6 all generate 6-point patterns, as we can see on the figure in the right, but the latter should be used because it is a rotation axis – the symbol will be
Finally, the Hermann–Mauguin symbol depends on the type of the group.

Groups without higher-order axes (axes of order three or more)

These groups may contain only two-fold axes, mirror planes, and/or an inversion center. These are the crystallographic point groups 1 and , 2, m, and , and 222,, and mm2. If the symbol contains three positions, then they denote symmetry elements in the x, y, z direction, respectively.

Groups with one higher-order axis

First position – primary direction – z direction, assigned to the higher-order axis.
Second position – symmetrically equivalent secondary directions, which are perpendicular to the z-axis. These can be 2, m, or
Third position – symmetrically equivalent tertiary directions, passing between secondary directions. These can be 2, m, or

These are the crystallographic groups 3, 32, 3m,, and , 4, 422, 4mm,, 2m,, and , and 6, 622, 6mm,, m2,, and . Analogously, symbols of non-crystallographic groups can be constructed. These groups can be arranged in the following table

Schoenflies	H–M symbol	3	4	5	6	7	8	9	10	11	12	...	∞
C_n	n	3	4	5	6	7	8	9	10	11	12	...	∞
C_nv	nm	3m		5m		7m		9m		11m		...	∞m
C_nv	nmm		4mm		6mm		8mm		10mm		12mm	...	∞m
S_2n												...
S_n												...
C_h												...
C_nh												...
D_n	n2	32		52		72		92		2		...	∞2
D_n	n22		422		622		822		22		22	...	∞2
D_nd												...	m
D_d	2m = m2		2m				2m				2m	...	m
D_h	2m = m2				m2				m2			...	m
D_nh												...	m

It can be noticed that in groups with odd-order axes n and ' the third position in symbol is always absent, because all n directions, perpendicular to higher-order axis, are symmetrically equivalent. For example, in the picture of a triangle all three mirror planes are equivalent – all of them pass through one vertex and the center of the opposite side.
For even-order axes n and ' there are secondary directions and tertiary directions. For example, in the picture of a regular hexagon one can distinguish two sets of mirror planes – three planes go through two opposite vertexes, and three other planes go through the centers of opposite sides. In this case any of two sets can be chosen as secondary directions, the rest set will be tertiary directions. Hence groups 2m, 2m, 2m, ... can be written as m2, m2, m2, ... . For symbols of point groups this order usually doesn't matter; however, it will be important for Hermann–Mauguin symbols of corresponding space groups, where secondary directions are directions of symmetry elements along unit cell translations b and c, while the tertiary directions correspond to the direction between unit cell translations b and c. For example, symbols Pm2 and P2m denote two different space groups. This also applies to symbols of space groups with odd-order axes 3 and. The perpendicular symmetry elements can go along unit cell translations b and c or between them. Space groups P321 and P312 are examples of the former and the latter cases, respectively.
The symbol of point group may be confusing; the corresponding Schoenflies symbol is D_3d, which means that the group consists of 3-fold axis, three perpendicular 2-fold axes, and 3 vertical diagonal planes passing between these 2-fold axes, so it seems that the group can be denoted as 32m or 3m2. However, one should remember that, unlike Schoenflies notation, the direction of a plane in a Hermann–Mauguin symbol is defined as the direction perpendicular to the plane, and in the D_3d group all mirror planes are perpendicular to 2-fold axes, so they should be written in the same position as. Second, these complexes generate an inversion center, which combining with the 3-fold rotation axis generates a rotoinversion axis.
Groups with n = ∞ are called limit groups or Curie groups.

Groups with several higher-order axes

These are the crystallographic groups of a cubic crystal system: 23, 432,, 3m, and. All of them contain four diagonal 3-fold axes. These axes are arranged as 3-fold axes in a cube, directed along its four space diagonals. These symbols are constructed the following way:

First position – symmetrically equivalent directions of the coordinate axes x, y, and z. They are equivalent due to the presence of diagonal 3-fold axes.
Second position – diagonal 3 or axes.
Third position – diagonal directions between any two of the three coordinate axes x, y, and z. These can be 2, m, or.

All Hermann–Mauguin symbols presented above are called full symbols. For many groups they can be simplified by omitting n-fold rotation axes in positions. This can be done if the rotation axis can be unambiguously obtained from the combination of symmetry elements presented in the symbol. For example, the short symbol for is mmm, for is mm, and for is mm. In groups containing one higher-order axis, this higher-order axis cannot be omitted. For example, symbols and can be simplified to 4/mmm and 6/mmm, but not to mmm; the short symbol for is m. The full and short symbols for all 32 crystallographic point groups are given in crystallographic point groups page.
Besides five cubic groups, there are two more non-crystallographic icosahedral groups and two limit groups. The Hermann–Mauguin symbols were not designed for non-crystallographic groups, so their symbols are rather nominal and based on similarity to symbols of the crystallographic groups of a cubic crystal system. Group I can be denoted as 235, 25, 532, 53. The possible short symbols for I_h are m, m, mm, m. The possible symbols for limit group K are ∞∞ or 2∞, and for K_h are ∞ or m or ∞∞m.