C3v E 2 C3 3 sv <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1 1 1 1 ... ..T ....T T.....T ..T.....T ....T.....T T.....T.....T A2 1 1 -1 ..T ... ..... .T..... ...T..... .....T..... .T.....T..... E 2 -1 0 TT. TT. TTTT. ..TTTT. TT..TTTT. TTTT..TTTT. ..TTTT..TTTT. Symmetry of Rotations and Cartesian products A1 p+d+2f+2g+2h+3i+3j+3k+4l+4m z, z^{2}, x(x^{2}−3y^{2}), z^{3}, xz(x^{2}−3y^{2}), z^{4}, xz^{2}(x^{2}−3y^{2}), z^{5}, x^{2}(x^{2}−3y^{2})^{2}−y^{2}(3x^{2}−y^{2})^{2}, xz^{3}(x^{2}−3y^{2}), z^{6} A2 R+f+g+h+2i+2j+2k+3l+3m R_{z}, y(3x^{2}−y^{2}), yz(3x^{2}−y^{2}), yz^{2}(3x^{2}−y^{2}), xy(x^{2}−3y^{2})(3x^{2}−y^{2}), yz^{3}(3x^{2}−y^{2}) E R+p+2d+2f+3g+4h+4i+5j+6k+6l+7m {R_{x}, R_{y}}, {x, y}, {x^{2}−y^{2}, xy}, {xz, yz}, {z(x^{2}−y^{2}), xyz}, {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}}, {xz^{3}, yz^{3}}, {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})}, {z((x^{2}−y^{2})^{2}−4x^{2}y^{2}), xyz(x^{2}−y^{2})}, {z^{3}(x^{2}−y^{2}), xyz^{3}}, {xz^{4}, yz^{4}}, {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})}, {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}}, {xz^{5}, yz^{5}} Notes: α The order of the C_{3v} point group is 6, and the order of the principal axis (C_{3}) is 3. The group has 3 irreducible representations. β The C_{3v} point group is isomorphic to D_{3}. It is also isomorphic to the Symmetric Group Sym(3). γ The C_{3v} point group is generated by two symmetry elements, C_{3} and any σ_{v}. Also, the group may be generated from any two σ_{v} planes. δ The group contains one set of symmetry planes σ_{v} intersecting in the principal (z) axis. The xz plane (but not the yz plane) is a member of that set. ε The lowest nonvanishing multipole moment in C_{3v} is 2 (dipole moment). ζ This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. η 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_{1v} | ||
C_{2v} | ||
C_{3} | C_{3v} | C_{3h} D_{3} D_{3h} D_{3d} |
C_{4v} | ||
C_{5v} |
This Character Table for the C_{3v} point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.