Schoenflies Notation - Point Groups

Point Groups

In three dimensions, there are infinite number of point groups, but all of them can be classified by several families.

• C_n (for cyclic) has an n-fold rotation axis.

• C_nh is C_n with the addition of a mirror (reflection) plane perpendicular to the axis of rotation (horizontal plane).
• C_nv is C_n with the addition of n mirror planes containing the axis of rotation (vertical planes).

• S_2n (for Spiegel, German for mirror) contains only a 2n-fold rotation-reflection axis. The index should be even because when n is odd an n-fold rotation-reflection axis is equivalent to combination of an n-fold rotation axis and a perpendicular plane, hence S_n = C_nh for odd n.
• C_ni has only a rotoinversion axis. These symbols are redundant, because any rotoinversion axis can be expressed as rotation-reflection axis, hence for odd n C_ni = S_2n and C_2ni = S_n = C_nh, and for even n C_2ni = S_2n. Only C_i is conventionally used, but in some texts you can see symbols like C_3i, C_5i.
• D_n (for dihedral, or two-sided) has an n-fold rotation axis plus n twofold axes perpendicular to that axis.

• D_nh has, in addition, a horizontal mirror plane and, as a consequence, also n vertical mirror planes each containing the n-fold axis and one of the twofold axes.
• D_nd has, in addition to the elements of D_n, n vertical mirror planes which pass between twofold axes (diagonal planes).

• T (the chiral tetrahedral group) has the rotation axes of a tetrahedron (three 2-fold axes and four 3-fold axes).

• T_d includes diagonal mirror planes (each diagonal plane contains only one twofold axis and passes between two other twofold axes, as in D_2d). This addition of diagonal planes results in three improper rotation operations S₄.
• T_h includes three horizontal mirror planes. Each plane contains two twofold axes and is perpendicular to the third twofold axis, which results in inversion center i.

• O (the chiral octahedral group) has the rotation axes of an octahedron or cube (three 4-fold axes, four 3-fold axes, and 6 diagonal 2-fold axes).

• O_h includes horizontal mirror planes and, as a consequence, vertical mirror planes. It contains also inversion center and improper rotation operations.

• I (the chiral icosahedral group) indicates that the group has the rotation axes of an icosahedron or dodecahedron (six 5-fold axes, ten 3-fold axes, and 15 2-fold axes).

• I_h includes horizontal mirror planes and contains also inversion center and improper rotation operations.

All groups that do not contain several higher-order axes (order 3 or more) can be arranged in a table:

The symbols that shouldn't be used are marked with maroon color.

In crystallography, due to the crystallographic restriction theorem, n is restricted to the values of 1, 2, 3, 4, or 6. The noncrystallographic groups are shown with grayed backgrounds. D_4d and D_6d are also forbidden because they contain improper rotations with n = 8 and 12 respectively. The 27 point groups in the table plus T, T_d, T_h, O and O_h constitute 32 crystallographic point groups.

Groups with n = ∞ are called limit groups or Curie groups. There are two more limit groups, not listed in the table: K (for Kugel, German for ball, sphere), the group of all rotations in 3-dimesional space; and K_h, the group of all rotations and reflections. In mathematics and theoretical physics they are known respectively as the special orthogonal group and the orthogonal group in three-dimensional space, with the symbols SO(3) and O(3).