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How are point groups different from space groups in crystallography?

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A rough, translucent yellow crystal of elemental sulfur displayed against a solid black background. — crystal A crystal form of elemental sulfur. Sulfur has an atomic weight of about 32.06 atomic mass units (amu), which means that its molar mass is 32.06 grams.

In crystallography, the primary difference between point groups and space groups lies in the inclusion of translational symmetry.12

Point Groups

Point groups, also known as crystal classes, describe the symmetry of a crystal based solely on operations that leave at least one point fixed. These operations include:12
* Rotation about an axis
* Reflection in a plane
* Inversion about a center
* Sequential rotation and inversion (rotary inversion)

There are exactly 32 distinct point groups, which can be determined by observing a crystal's external morphology.12

Space Groups

Space groups incorporate the same point operations as point groups but add translational changes—displacing the structure along a crystallographic axis so that atoms are replaced by identical neighbors. By including these translational elements, there are 230 possible space groups. While point groups describe external shape, space groups provide a definitive categorization of the internal atomic arrangement, typically determined through methods like X-ray crystallography.132

Would you like to know more about how these symmetries relate to the 14 Bravais lattices?