Lattice character directions of 32 original point groups

Crystallographic system	Point Groups	Direction 1	Direction 2	Direction 3
Cubic	$23, m \overline{3}, 432, \overline{4} 3 m$ $m \overline{3} m$	$\mathbf{a}$ $\mathbf{b}$ $\mathbf{c}$	$\mathbf{a}+\mathbf{b}+\mathbf{c}$	$\mathbf{a}+\mathbf{b}$
Hexagonal	$6, \quad \overline{6}, \quad 6 / m, \quad 622$ $6 m m, \quad \overline{6} m 2$ $6 / \mathrm{mmm}$	$\mathbf{c}$	$\mathbf{a}$ $\mathbf{b}$	$\mathbf{a}+\mathbf{2 b} / \mathbf{2 a}+\mathbf{b}$
Tetragonal	$4, \overline{4}, 4 / m, 422$ $4 m m, \quad \overline{4} 2 m$ $4 / \mathrm{mmm}$	$\mathbf{c}$	$\mathbf{a}$ $\mathbf{b}$	$\mathbf{a}+\mathbf{b}$
Trigonal	$3, \overline{3}, 32,3 m, \overline{3} m$	$\mathbf{c}$	$\mathbf{a}$ $\mathbf{b}$	NA
Orthorhombic	$222, m m 2, m m m$	$\mathbf{a}$	$\mathbf{b}$	$\mathbf{c}$
Monoclinic	$2, m, 2 / m$	$\mathbf{b}$ (unique axis b)	NA	NA
Triclinic	$\overline{1}$	NA	NA	NA

Notes:

The letter "NA" (not available) in the third to the fifth column denotes no symmetries along these

character directions.

"/" in the third to the fifth column denotes there are two or more equivalent character directions.
The character directions for grey MPGs and black-white MPGs obey similar rules to original MPGs.

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