List of space groups

There are 230 space groups in three dimensions, given by a number index, and a full name in Hermann–Mauguin notation, and a short name (international short symbol). The long names are given with spaces for readability. The groups each have a point group of the unit cell.

Symbols

In Hermann–Mauguin notation, space groups are named by a symbol combining the point group identifier with the uppercase letters describing the lattice type. Translations within the lattice in the form of screw axes and glide planes are also noted, giving a complete crystallographic space group.

These are the Bravais lattices in three dimensions:

P primitive

I body-centered (from the German Innenzentriert)

F face-centered (from the German Flächenzentriert)

S base-centered (from the German Seitenflächenzentriert), or specifically:

A centered on A faces only

B centered on B faces only

C centered on C faces only

R rhombohedral

A reflection plane m within the point groups can be replaced by a glide plane, labeled as a, b, or c depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is along a quarter of either a face or space diagonal of the unit cell. The d glide is often called the diamond glide plane as it features in the diamond structure.

a

{\displaystyle a}

,

b

{\displaystyle b}

, or

c

{\displaystyle c}

: glide translation along half the lattice vector of this face

n

{\displaystyle n}

: glide translation along half the diagonal of this face

d

{\displaystyle d}

: glide planes with translation along a quarter of a face diagonal

e

{\displaystyle e}

: two glides with the same glide plane and translation along two (different) half-lattice vectors.

A gyration point can be replaced by a screw axis denoted by a number, n, where the angle of rotation is

360

∘

n

{\displaystyle \color {Black}{\tfrac {360^{\circ }}{n}}}

.