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The finite orthogonal group obtained by quotienting a crystal space group by its translation subgroup. In three dimensions there are 32 crystallographic point-group types, constrained by preservation of a lattice. It describes macroscopic rotational and reflection symmetry, not the fractional translations of the full space group.

Second online edition (2016), International Union of crystallography. Authoritative tables of crystallographic point groups and space groups. it.iucr.org/Ac/ doi.org/10.1107/97809553602060000114

A space group whose point-group operations cannot all be represented without essential fractional translations for any origin. screw axes and glide planes are characteristic examples. Its point group alone omits these translational data.

A Bravais lattice is the translation set $\Lambda =\{n_1{\mathbf{a}}_1+n_2{\mathbf{a}}_2+n_3{\mathbf{a}}_3:n_i\in \mathbb{Z}\}$ generated by primitive lattice translation vectors. Translating by any member of $\Lambda$ leaves its surroundings equivalent. A crystal structure requires both this lattice and an atomic basis attached to every lattice point. The basis can lower the symmetry of the bare point lattice; a familiar-looking geometrical arrangement of atoms is not necessarily itself a Bravais lattice. For example, the diamond cubic structure is a face-centred cubic lattice with an identical-atom basis at fractional positions $(0,0,0)$ and $(1/4,1/4,1/4)$ in conventional cubic coordinates. The displacement between those two basis atoms is not itself a lattice translation.

The seven three-dimensional crystal systems are triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic; the distinct translation types give 14 Bravais lattices. A conventional cell can contain several lattice points, while a primitive unit cell contains one. Simple cubic primitive lattice translation vectors can be chosen along the cube edges; for face-centred cubic they can be $(a/2)(0,1,1)$, $(a/2)(1,0,1)$ and $(a/2)(1,1,0)$. Their primitive volume is $a^3/4$, explaining the four lattice points per conventional face-centred cell.

A crystallographic point group comprises the orthogonal parts of the crystal's spatial symmetry operations. It constrains possible rotations, reflections, inversion and their combinations, while forgetting translations. Lattice preservation restricts proper rotational orders in three dimensions to $1,2,3,4,6$; a fivefold axis cannot be a symmetry of a periodic three-dimensional lattice. One quick argument restricts a rotation's integral lattice representation through $1+2\cos \theta$, an integer trace, producing these orders. There are 32 crystallographic point-group types. Examples are the cubic full octahedral group $O_h$ with Hermann–Mauguin symbol $m\bar{3}m$, the tetrahedral class $T_d$ with symbol $\bar{4}3m$, and the polar tetragonal class $C_{4v}$ with symbol $4mm$.

The space group retains the full isometry. In Seitz notation $\{R\mid \mathbf{t}\}$ acts by $\mathbf{x}\mapsto R\mathbf{x}+\mathbf{t}$, and multiplication isIts translation subgroup is normal and the quotient is the crystallographic point group. A screw axis combines rotation with translation along the axis; for example, applying a $2_1$ screw twice gives one whole lattice translation. A glide plane combines reflection with a translation parallel to the plane; applying the glide twice likewise gives a translation. These operations can preserve the complete crystal without their rotation or reflection alone preserving it.

A symmorphic space group admits an origin and representatives for which the point-group operations have no translation component. A nonsymmorphic space group has essential fractional translations that cannot all be removed by moving the origin. There are 230 three-dimensional space-group types, distinguished more finely than their point groups. These counts refer to ordinary, nonmagnetic periodic space groups; magnetic and aperiodic generalizations are different classifications.^[ref]

For concrete comparisons, a monatomic simple-cubic crystal has space group $Pm\bar{3}m$ and point group $m\bar{3}m$. Rock salt has a face-centred cubic lattice, a two-species basis and symmorphic space group $Fm\bar{3}m$. Diamond has the same cubic point group and translation-lattice type, but its basis leads to nonsymmorphic $Fd\bar{3}m$. Replacing the two diamond basis sites by distinct species gives the zinc blende structure, exemplified by GaAs, with $F\bar{4}3m$ and point group $\bar{4}3m$; inversion that would interchange the species is no longer a symmetry. The lattice specifies translations, the point group specifies the linear symmetry classes, and the space group specifies their complete spatial implementation. These examples show why none alone uniquely describes an atomic structure.

Patterson maps give interatomic displacement vectors rather than absolute positions. The three strong half-cell vectors are consistent with a face-centered four-point Se substructure. One permissible coordinate set isEach nonzero vector occurs four times among the twelve ordered Se-Se pairs, explaining equal weights. These positions form a valid four-site orbit of the specified space group in a suitable origin convention.

A common translation of all four sites is invisible to a Patterson map; absolute coordinates are therefore not uniquely fixed by these three peaks. Subsequent phases are stated in the origin convention above. The remaining atoms need not have face-centered translational symmetry merely because the dominant Se substructure does.

A space group for which an origin can be chosen so that representatives of all point-group operations have zero translation part. It is a semidirect product of its lattice translations and point group. Presence of centring translations in a conventional cell does not alone make a group nonsymmorphic.