C3h E 2 C3 sh 2 S3 <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A' 1 1 1 1 ..T ... ....T TT..... ........T ....TT..... TT..........T A" 1 1 -1 -1 ... ..T ..... ......T ..TT..... ..........T ......TT..... E' * 2 -1 2 -1 ... TT. TT... ....TT. TT..TT... TT......TT. ....TT..TT... E" * 2 -1 -2 1 TT. ... ..TT. ..TT... ......TT. ..TT..TT... ..TT......TT. Symmetry of Rotations and Cartesian products A' R+d+2f+g+2h+3i+2j+3k+4l+3m R_{z}, z^{2}, x(x^{2}−3y^{2}), y(3x^{2}−y^{2}), z^{4}, xz^{2}(x^{2}−3y^{2}), yz^{2}(3x^{2}−y^{2}), x^{2}(x^{2}−3y^{2})^{2}−y^{2}(3x^{2}−y^{2})^{2}, xy(x^{2}−3y^{2})(3x^{2}−y^{2}), z^{6} A" p+f+2g+h+2i+3j+2k+3l+4m z, z^{3}, xz(x^{2}−3y^{2}), yz(3x^{2}−y^{2}), z^{5}, xz^{3}(x^{2}−3y^{2}), yz^{3}(3x^{2}−y^{2}) E' p+d+f+2g+2h+2i+3j+3k+3l+4m {x, y}, {x^{2}−y^{2}, xy}, {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}}, {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^{4}, yz^{4}}, {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}} E" R+d+f+g+2h+2i+2j+3k+3l+3m {R_{x}, R_{y}}, {xz, yz}, {z(x^{2}−y^{2}), xyz}, {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}}, {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^{5}, yz^{5}} Notes: α The order of the C_{3h} point group is 6, and the order of the principal axis (S_{3}) is 6. The group has 4 irreducible representations. β The C_{3h} point group could also be named S_{3}, as it contains the S_{3} axis as its only symmetry element. Another rare designation is C_{6i} because the S_{3} axis is identical to a roto-inversion axis of order 6. γ The C_{3h} point group is isomorphic to C_{6} and S_{6}. δ The C_{3h} point group is generated by one single symmetry element, S_{3}. Therefore, it is a cyclic group. The canonical choice, however, is to use redundant generators: C_{3} and σ_{h}. ε The lowest nonvanishing multipole moment in C_{3h} is 4 (quadrupole moment). ζ This is an Abelian point group (the commutative law holds between all symmetry operations). The C_{3h} group is Abelian because it contains only one symmetry element, all the powers of which necessarily commute (sufficient condition). In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional. η Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters. These 4 cases have been combined into 2 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 2 pairs of left and right rotations have been combined into one two-membered pseudo-class each. θ The 2 reducible “E” representations almost behave like true irreducible representations. Their norm, however, is twice the group order. Therefore, they have been marked with an asterisk in the table. This is essential when trying to decompose a reducible representation into “irreducible” ones using the familiar projection formula. ι All characters are integers because the order of the principal axis is 1,2,3,4 or 6. This implies that the point group corresponds to a constructible polygon which can be used for tiling the plane. 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_{s} | ||
C_{2h} | ||
C_{3} C_{3v} | C_{3h} | D_{3} D_{3h} D_{3d} |
C_{4h} | ||
C_{5h} |
This Character Table for the C_{3h} point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.