Lecture 7 • Autumn 2026

Deriving Hermann-Mauguin Notations

Understanding standard crystallographic point group nomenclatures using proper rotations, improper rotations, and directional hierarchies.

Rules & Conventions

Hermann-Mauguin Scheme

Preferred by crystallographers, this notation can be easily extended to include translational symmetry elements and specifies the directions of the symmetry axes.

Base Operations

  • Proper rotations: Described by the order of rotation: e.g., $n$-fold axis, represented as $n$.
  • Reflections: Described by the symbol $m$. The direction is taken as normal to $m$.
  • Inversion: Described by the symbol $i$ or $\bar{1}$.
  • Roto-inversion: Described by the symbol $\bar{n}$ (where $n$ is the order of rotation).

Positional Hierarchy

H-M notations consist of at most three components, each referring to a different direction.

  1. First position: Rotation axis of highest order.
  2. Second position: Symmetrically equivalent secondary directions.
  3. Third position: Symmetrically equivalent tertiary directions passing between secondary directions.

Note: The position of $m$ indicates the direction of normal to the mirror plane.

Interactive H-M Derivation Tool

Select a point group target and construct its symmetry step-by-step. Enable secondary/tertiary positions to see cross-generated symmetry elements sequentially map the points.

Target Group

Cyan shapes: symmetry elements.
Black points: target pattern.

Your Construction

Live Build

Operations duplicate iteratively.
Blue points mapped live.

Construct H-M Notation

Key Takeaways

In summary..

Full symbol in H-M notations is built by the following process:

  • Identify the symmetry axes that might work as primary, secondary or tertiary directions
  • $n$ = proper rotation, $\bar{n}$ = improper rotation (roto-inversion) and $m$ = mirror
  • Primary, secondary and tertiary directions must not be equivalent by symmetry
  • Symmetry axes are arranged in decreasing order (except for $23$ and $\frac{2}{m}\bar{3}$ in tetrahedral groups to avoid confusion with dihedral groups $32$ ($D_3$) and $\bar{3}\frac{2}{m}$ ($D_{3d}$))
  • Secondary direction exists when a symmetry element in a direction other than the primary direction exists
  • Tertiary direction exists when a symmetry element other than that described by primary and secondary directions and one that is not equivalent by symmetry to primary and secondary directions exists
  • Inversion never appears in any group symbol
  • $\bar{n}$ does not necessarily imply that an $n$ axis and a mirror exist independently
  • For $n$ = odd, $\bar{n}$ implies presence of both $n$ and $i$

Schoenflies $\leftrightarrow$ Hermann-Mauguin

Comprehensive mapping table comparing Schoenflies notation with rigorous, full Hermann-Mauguin point group symbols.

S. No. Schoenflies H-M S. No. Schoenflies H-M S. No. Schoenflies H-M S. No. Schoenflies H-M
1 $C_1 = E$ 1 9 $S_4$ $\bar{4}$ 17 $C_{6h}$ $\frac{6}{m}$ 25 $D_{3h}$ $\bar{6}m2$
2 $C_2 = D_1$ 2 10 $S_6$ $\bar{3}$ 18 $D_2$ 222 26 $D_{4h}$ $\frac{4}{m}\frac{2}{m}\frac{2}{m}$
3 $C_3$ 3 11 $C_{2v}$ $2mm = mm2$ 19 $D_3$ 32 27 $D_{6h}$ $\frac{6}{m}\frac{2}{m}\frac{2}{m}$
4 $C_4$ 4 12 $C_{3v}$ $3m$ 20 $D_4$ 422 28 $T$ 23
5 $C_6$ 6 13 $C_{4v}$ $4mm$ 21 $D_6$ 622 29 $T_h$ $\frac{2}{m}\bar{3}$
6 $C_i$ $\bar{1}$ 14 $C_{6v}$ $6mm$ 22 $D_{2d}$ $\bar{4}2m$ 30 $T_d$ $\bar{4}3m$
7 $S_1 = C_s$ $\bar{2} = m$ 15 $C_{2h}$ $\frac{2}{m}$ 23 $D_{3d}$ $\bar{3}\frac{2}{m}$ 31 $O$ 432
8 $S_3 = C_{3h}$ $\bar{6} = \frac{3}{m}$ 16 $C_{4h}$ $\frac{4}{m}$ 24 $D_{2h}$ $\frac{2}{m}\frac{2}{m}\frac{2}{m}$ 32 $O_h$ $\frac{4}{m}\bar{3}\frac{2}{m}$