Crystallographic point group

A crystallographic point group is a group of rotations, reflections, inversions, or rotoinversions that fixes at least one point and is compatible with lattice periodicity. In three dimensions, exactly 32 such groups are possible.

Symmetry operation: A transformation that maps an object onto an indistinguishable arrangement. Point groups are built from transformations that preserve a crystal's appearance.
Group theory: The mathematical study of sets equipped with an associative operation, an identity, and inverses. Point-group operations form groups under composition.
Crystal system: One of seven categories of crystal lattices classified by unit-cell geometry and symmetry. Each crystal system permits a characteristic subset of point groups.
Space group: A group of crystal symmetries that includes both point operations and translations. Unlike point groups, space groups include translations and describe full periodic structures.
Crystal lattice: An infinite periodic arrangement of points that represents a crystal's translational structure. Its periodicity restricts which rotations can preserve a crystal.
Group action: A way for group elements to act as transformations on a set while respecting group composition. A point group acts on directions and positions around its fixed point.
Crystal class: A symmetry classification of crystals based on their crystallographic point group. The 32 point groups are the 32 crystal classes.
Molecular point group: A group of symmetry operations that leaves a molecule unchanged while fixing a point. Molecular point groups need not satisfy the rotational restrictions imposed by lattice periodicity.
Rotation symmetry: A symmetry in which rotation about an axis leaves an object unchanged. Allowed rotational axes are among the defining operations of point groups.
Symmetry axis: An axis about which a rotation maps an object onto itself. Its order and orientation help determine a point group's structure.

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