TheSchoenflies (orSchönflies)notation, named after theGerman mathematicianArthur Moritz Schoenflies, is a notation primarily used to specifypoint groups in three dimensions. Because a point group alone is completely adequate to describe thesymmetry of a molecule, the notation is often sufficient and commonly used forspectroscopy. However, incrystallography, there is additionaltranslational symmetry, and point groups are not enough to describe the full symmetry of crystals, so the fullspace group is usually used instead. The naming of full space groups usually follows another common convention, theHermann–Mauguin notation, also known as the international notation.

Although Schoenflies notation without superscripts is a pure point group notation, optionally, superscripts can be added to further specify individual space groups. However, for space groups, the connection to the underlyingsymmetry elements is much more clear in Hermann–Mauguin notation, so the latter notation is usually preferred for space groups.

Symmetry elements are denoted byi for centers of inversion,C for proper rotation axes,σ for mirror planes, andS for improper rotation axes (rotation-reflection axes).C andS are usually followed by a subscript number (abstractly denotedn) denoting the order of rotation possible.

By convention, the axis of proper rotation of greatest order is defined as the principal axis. All other symmetry elements are described in relation to it. A vertical mirror plane (containing the principal axis) is denotedσ_v; a horizontal mirror plane (perpendicular to the principal axis) is denotedσ_h.

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

• C_n (forcyclic) has ann-fold rotation axis.
  • C_nh isC_n with the addition of a mirror (reflection) plane perpendicular to the axis of rotation (horizontal plane).
  • C_nv isC_n with the addition ofn mirror planes containing the axis of rotation (vertical planes).
• C_s denotes a group with only mirror plane (forSpiegel, German for mirror) and no other symmetry elements.
• S_n (forSpiegel, German formirror) contains only an-foldrotation-reflection axis. The index,n, should be even because when it is odd ann-fold rotation-reflection axis is equivalent to a combination of ann-fold rotation axis and a perpendicular plane, henceS_n =C_nh for oddn.
• C_ni has only arotoinversion axis. This notation is rarely used because any rotoinversion axis can be expressed instead as rotation-reflection axis: For oddn,C_ni =S_2n andC_2ni =S_n =C_nh, and for evenn,C_2ni =S_2n. Only the notationC_i (meaningC_1i) is commonly used, and some sources writeC_3i,C_5i etc.
• D_n (fordihedral, or two-sided) has ann-fold rotation axis plusn twofold axes perpendicular to that axis.
  • D_nh has, in addition, a horizontal mirror plane and, as a consequence, alson vertical mirror planes each containing then-fold axis and one of the twofold axes.
  • D_nd has, in addition to the elements ofD_n,n vertical mirror planes which pass between twofold axes (diagonal planes).
• T (the chiraltetrahedral 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 inD_2d). This addition of diagonal planes results in three improper rotation operationsS₄.
  • 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 centeri.
• O (the chiraloctahedral group) has the rotation axes of an octahedron orcube (three 4-fold axes, four 3-fold axes, and six 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 chiralicosahedral group) indicates that the group has the rotation axes of an icosahedron ordodecahedron (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 more than one higher-order axis (order 3 or more) can be arranged as shown in a table below; symbols in red are rarely used.

In crystallography, due to thecrystallographic 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 andD_6d are also forbidden because they containimproper rotations withn = 8 and 12 respectively. The 27 point groups in the table plusT,T_d,T_h,O andO_h constitute 32crystallographic point groups.

Groups withn = ∞ are called limit groups orCurie groups. There are two more limit groups, not listed in the table:K (forKugel, German for ball, sphere), the group of all rotations in 3-dimensional space; andK_h, the group of all rotations and reflections. In mathematics and theoretical physics they are known respectively as thespecial orthogonal group and theorthogonal group in three-dimensional space, with the symbols SO(3) and O(3).

Thespace groups with given point group are numbered by 1, 2, 3, ... (in the same order as their international number) and this number is added as a superscript to the Schönflies symbol for the corresponding point group. For example, groups numbers 3 to 5 whose point group isC₂ have Schönflies symbolsC¹
₂,C²
₂,C³
₂.

While in case of point groups, Schönflies symbol defines the symmetry elements of group unambiguously, the additional superscript for space group doesn't have any information about translational symmetry of space group (lattice centering, translational components of axes and planes), hence one needs to refer to special tables, containing information about correspondence between Schönflies andHermann–Mauguin notation. Such table is given inList of space groups page.

• Crystallographic point group
• Point groups in three dimensions
• List of spherical symmetry groups

• Flurry, R. L.,Symmetry Groups : Theory and Chemical Applications. Prentice-Hall, 1980.ISBN 978-0-13-880013-0 LCCN: 79-18729
• Cotton, F. A.,Chemical Applications of Group Theory, John Wiley & Sons: New York, 1990.ISBN 0-471-51094-7
• Harris, D., Bertolucci, M.,Symmetry and Spectroscopy. New York, Dover Publications, 1989.

• Symmetry @ Otterbein

• Symmetry
• Spectroscopy
• Crystallography

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• Short description matches Wikidata