Space Group Symbols of Non-Orthorhombic Systems
In orthorhombic space groups, after the centring symbol, Hermann-Mauguin notation gives three symbols, corresponding to symmetry elements with unique axis \(a\), \(b\) and \(c\) respectively. For other crystal systems, this process is slightly more complicated. The table below details the unique axes used for space group symbols for each of the crystal systems.
| Crystal System | First Symbol | Second Symbol | Third Symbol |
| Triclinic | 1 or \(\bar{1}\) | – | – |
| Monoclinic | 1 | \(\mathbf{b}\) | 1 |
| Orthorhombic | \(\mathbf{a}\) | \(\mathbf{b}\) | \(\mathbf{c}\) |
| Tetragonal | \(\mathbf{c}\) | \(\mathbf{a}\) | \(\mathbf{a}-\mathbf{b}\) |
| Trigonal \(H\) | \(\mathbf{c}\) | \(\mathbf{a}\) | \(\mathbf{a}-\mathbf{b}\) |
| Trigonal \(R\) | \(\mathbf{c}_{H}\) or \(\mathbf{a}_{R} + \mathbf{b}_{R} + \mathbf{c}_{R}\) | \(\mathbf{a}_{H}\) or \(\mathbf{a}_{R}-\mathbf{b}_{R}\) | – |
| Hexagonal | \(\mathbf{c}\) | \(\mathbf{a}\) | \(\mathbf{a}-\mathbf{b}\) |
| Cubic | \(\mathbf{c}\) | \(\mathbf{a} + \mathbf{b} + \mathbf{c}\) | \(\mathbf{a}-\mathbf{b}\) |
By cross-referencing each symbol in a space group symbol with the table above, the unique axes of the symmetry elements can be deduced. Multiple examples will be given in the next few pages.
Note that different settings are possible for the trigonal crystal system, and the vectors used to define symmetry elements in either setting are identical (they have the same direction within the lattice). However, these are formed from the base hexagonal and rhombohedral unit cell vectors differently.
The following pages will deal with each case in order to ensure complete understanding of space group symbols.