C1h E sh <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A' 1 1 ..T TT. TT..T TT..TT. TT..TT..T TT..TT..TT. TT..TT..TT..T A" 1 -1 TT. ..T ..TT. ..TT..T ..TT..TT. ..TT..TT..T ..TT..TT..TT. Symmetry of Rotations and Cartesian products A' R+2p+3d+4f+5g+6h+7i+8j+9k+10l+11m R_{z}, x, y, x^{2}−y^{2}, xy, z^{2}, x(x^{2}−3y^{2}), y(3x^{2}−y^{2}), xz^{2}, yz^{2}, (x^{2}−y^{2})^{2}−4x^{2}y^{2}, xy(x^{2}−y^{2}), z^{2}(x^{2}−y^{2}), xyz^{2}, z^{4}, x(x^{2}−(5+2√5)y^{2})(x^{2}−(5−2√5)y^{2}), y((5+2√5)x^{2}−y^{2})((5−2√5)x^{2}−y^{2}), xz^{2}(x^{2}−3y^{2}), yz^{2}(3x^{2}−y^{2}), xz^{4}, yz^{4}, x^{2}(x^{2}−3y^{2})^{2}−y^{2}(3x^{2}−y^{2})^{2}, xy(x^{2}−3y^{2})(3x^{2}−y^{2}), z^{2}((x^{2}−y^{2})^{2}−4x^{2}y^{2}), xyz^{2}(x^{2}−y^{2}), z^{4}(x^{2}−y^{2}), xyz^{4}, z^{6} A" 2R+p+2d+3f+4g+5h+6i+7j+8k+9l+10m R_{x}, R_{y}, z, xz, yz, z(x^{2}−y^{2}), xyz, z^{3}, xz(x^{2}−3y^{2}), yz(3x^{2}−y^{2}), xz^{3}, yz^{3}, z((x^{2}−y^{2})^{2}−4x^{2}y^{2}), xyz(x^{2}−y^{2}), z^{3}(x^{2}−y^{2}), xyz^{3}, z^{5}, xz(x^{2}−(5+2√5)y^{2})(x^{2}−(5−2√5)y^{2}), yz((5+2√5)x^{2}−y^{2})((5−2√5)x^{2}−y^{2}), xz^{3}(x^{2}−3y^{2}), yz^{3}(3x^{2}−y^{2}), xz^{5}, yz^{5} Notes: α The order of the C_{1h} point group is 2, and the order of the principal axis (σ_{h}) is 2. The group has 2 irreducible representations. β The C_{1h} point group is usually referred to as C_{s}. It could also be named S_{1} or C_{2i}. γ The C_{s} point group is isomorphic to C_{2} and C_{i}, and also to the Symmetric Group Sym(2). δ The C_{s} point group is generated by one single symmetry element, σ_{h}. Therefore, it is a cyclic group. ε The lowest nonvanishing multipole moment in C_{1h} is 2 (dipole moment). ζ This is an Abelian point group (the commutative law holds between all symmetry operations). The C_{s} group is Abelian because it meets two conditions, each of one alone would have been sufficient: It contains only one symmetry element (σ_{h}), and there is no axis of order 3 or higher. In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional. η There are no symmetry elements of an order higher than 2 in this group. The symmetry-adapted Cartesian products in the table above are needlessly complicated; rather, any simple product will do. θ C_{s} is often termed a nonaxial group, although the mirror plane does have a preferred direction. This term is also used for C_{1} and C_{i}, and occasionally for C_{2}. ι All characters are integers because the order of the principal axis is 1,2,3,4 or 6. Such point groups are also referred to as “crystallographic point groups”, as they are compatible with periodic lattice symmetry. There are exactly 32 such groups: C_{1},C_{s},C_{i},C_{2},C_{2h},C_{2v},C_{3},C_{3h},C_{3v},C_{4},C_{4h},C_{4v},C_{6},C_{6h},C_{6v},D_{2},D_{2d},D_{2h},D_{3},D_{3d},D_{3h},D_{4},D_{4h},D_{6},D_{6h},S_{4},S_{6},T,T_{d},T_{h},O,O_{h}.
C_{1} C_{1v} | C_{s} | D_{1} D_{1h} D_{1d} |
C_{2h} | ||
C_{3h} |
This Character Table for the C_{s} point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.