Space Groups

A space group is the symmetry group of a repeating pattern in space. The elements of the space group are the rigid transformations that leave the pattern unchanged. The point groups are $\subset$ of the space groups. The set of all point groups and translations form the space groups.

Symmorphic

A space group is called ‘symmorphic’ if, apart from lattice rotations, all generating symmetry operations leave one common point fixed. Thus, only point-group operations are permitted: rotations, reflections, inversions, and rotoinversions.

Herman-Maugin Space Group Symbol

The HM space group symbol can be derived from the symmetry elements present using the following logic.

The first letter (lattice descriptor) identifies the centering of the lattice

• P: Primitive
• I : Body-centered
• F: Face-centered
• C: C-centered
• B: B-centered
• A: A-centered

The next three symbols denote the symmetry elements present in certain directions

Cystal System	Symmetry Direction	—	—
---	Primary	Secondary	Tertiary
Triclinic	None
Monoclinic	$[010]$
Orthorhombic	$[100]$	$[010]$	$[001]$
Tetragonal	$[001]$	$[100]/[010]$	$[110]$
Hexagonal	$[001]$	$[100]/[010]$	$[120]/[1\bar{1}0]$
Cubic	$[100]/[010]/[001]$	$[111]$	$[110]$

• $[100]$ - Axis parallel or plane-perpendicular to x-axis
• $[010]$ - Axis parallel or plane perpendicular to y-axis
• $[001]$ - Axis parallel or plane perpendicular to z-axis
• $[110]$ - Axis parallel or plane perpendicular to line running $45$ degrees to x and y axes
• $[1\bar{1}0]$ - Axis parallel or plane perpendicular to long face diagonal of ab face of a hexagonal cell
• $[111]$ - Axis parallel or plane perpendicular to the body diagonal