These 5 point groups have 7 corresponding space groups (denoted by R) assigned to the rhombohedral lattice system and 18 corresponding space groups (denoted by P) assigned to the hexagonal lattice system. The trigonal crystal system consists of the 5 point groups that have a single three-fold rotation axis, which includes space groups 143 to 167. A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to a lattice system (see table in Crystal system#Crystal classes). The hexagonal crystal family consists of two crystal systems: trigonal and hexagonal. However, such a description is rarely used. The rhombohedral unit cell for the hexagonal Bravais lattice is the D-centered cell, consisting of two additional lattice points which occupy one body diagonal of the unit cell with coordinates ( 1⁄ 3, 1⁄ 3, 1⁄ 3) and ( 2⁄ 3, 2⁄ 3, 2⁄ 3). However, the rhombohedral axes are often shown (for the rhombohedral lattice) in textbooks because this cell reveals 3m symmetry of crystal lattice. In practice, the hexagonal description is more commonly used because it is easier to deal with a coordinate system with two 90° angles. This is a unit cell with parameters a = b = c α = β = γ ≠ 90°. The unit cell is a rhombohedron (which gives the name for the rhombohedral lattice). The Bravais lattices in the hexagonal crystal family can also be described by rhombohedral axes.
In either case, there are 3 lattice points per unit cell in total and the lattice is non-primitive. In the usual so-called obverse setting, the additional lattice points are at coordinates ( 2⁄ 3, 1⁄ 3, 1⁄ 3) and ( 1⁄ 3, 2⁄ 3, 2⁄ 3), whereas in the alternative reverse setting they are at the coordinates ( 1⁄ 3, 2⁄ 3, 1⁄ 3) and ( 2⁄ 3, 1⁄ 3, 2⁄ 3). There are two ways to do this, which can be thought of as two notations which represent the same structure.
The hexagonal unit cell for the rhombohedral Bravais lattice is the R-centered cell, consisting of two additional lattice points which occupy one body diagonal of the unit cell. In the hexagonal family, the crystal is conventionally described by a right rhombic prism unit cell with two equal axes ( a by a), an included angle of 120° ( γ) and a height ( c, which can be different from a) perpendicular to the two base axes. Relation between the two settings for the rhombohedral lattice Hexagonal crystal family