D2d E 2 S4 C2 2 C2' 2 sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1 1 1 1 1 1 ... ... ....T ...T... T.......T .......T... ....T.......T A2 1 1 1 -1 -1 ..T ... ..... ..T.... .T....... ......T.... .....T....... B1 1 -1 1 1 -1 ... ... T.... ....... ....T.... ...T....... T.......T.... B2 1 -1 1 -1 1 ... ..T .T... ......T .....T... ..T.......T .T.......T... E 2 0 -2 0 0 TT. TT. ..TT. TT..TT. ..TT..TT. TT..TT..TT. ..TT..TT..TT. Symmetry of Rotations and Cartesian products A1 d+f+2g+h+2i+2j+3k+2l+3m z^{2}, xyz, (x^{2}−y^{2})^{2}−4x^{2}y^{2}, z^{4}, xyz^{3}, z^{2}((x^{2}−y^{2})^{2}−4x^{2}y^{2}), z^{6} A2 R+f+g+h+i+2j+2k+2l+2m R_{z}, z(x^{2}−y^{2}), xy(x^{2}−y^{2}), z^{3}(x^{2}−y^{2}), xyz^{2}(x^{2}−y^{2}) B1 d+g+h+2i+j+2k+2l+3m x^{2}−y^{2}, z^{2}(x^{2}−y^{2}), xyz(x^{2}−y^{2}), x^{2}(x^{2}−3y^{2})^{2}−y^{2}(3x^{2}−y^{2})^{2}, z^{4}(x^{2}−y^{2}) B2 p+d+f+g+2h+2i+2j+2k+3l+3m z, xy, z^{3}, xyz^{2}, z((x^{2}−y^{2})^{2}−4x^{2}y^{2}), z^{5}, xy(x^{2}−3y^{2})(3x^{2}−y^{2}), xyz^{4} E R+p+d+2f+2g+3h+3i+4j+4k+5l+5m {R_{x}, R_{y}}, {x, y}, {xz, yz}, {x(x^{2}−3y^{2}), y(3x^{2}−y^{2})}, {xz^{2}, yz^{2}}, {xz(x^{2}−3y^{2}), yz(3x^{2}−y^{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})}, {xz^{2}(x^{2}−3y^{2}), yz^{2}(3x^{2}−y^{2})}, {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})}, {xz^{3}(x^{2}−3y^{2}), yz^{3}(3x^{2}−y^{2})}, {xz^{5}, yz^{5}} Notes: α The order of the D_{2d} point group is 8, and the order of the principal axis (S_{4}) is 4. The group has 5 irreducible representations. β The D_{2d} point group is also known as V_{d}. The letter V derives from German ‘Vierergruppe’ (group of four) for the Klein four-group, to which D_{2} is isomorphic. γ The D_{2d} point group is isomorphic to C_{4v} and D_{4}. δ The D_{2d} point group is generated by two symmetry elements, S_{4} and either a perpendicular C_{2}^{′} or a vertical σ_{d}. Also, the group may be generated from any C_{2}^{′} plus any σ_{d} plane. ε The group contains two C_{2}^{′} symmetry axes perpendicular to the principal (z) axis, which are by convention chosen as x and y. ζ The single σ_{d} set of symmetry planes contains neither the xz nor the yz planes; but it contains the median plane (x+y)z. η The lowest nonvanishing multipole moment in D_{2d} is 4 (quadrupole moment). θ This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. 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}.
D_{1d} | ||
C_{2} C_{2v} C_{2h} D_{2} D_{2h} | D_{2d} | C_{i} |
D_{3d} | ||
D_{4d} |
This Character Table for the D_{2d} point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.