D2 E C2 C2' C2" <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A 1 1 1 1 ... ... T...T ...T... T...T...T ...T...T... T...T...T...T B1 1 1 -1 -1 ..T ..T .T... ..T...T .T...T... ..T...T...T .T...T...T... B2 1 -1 -1 1 .T. .T. ..T.. .T...T. ..T...T.. .T...T...T. ..T...T...T.. B3 1 -1 1 -1 T.. T.. ...T. T...T.. ...T...T. T...T...T.. ...T...T...T. Symmetry of Rotations and Cartesian products A 2d+f+3g+2h+4i+3j+5k+4l+6m x^{2}−y^{2}, z^{2}, xyz, (x^{2}−y^{2})^{2}−4x^{2}y^{2}, z^{2}(x^{2}−y^{2}), z^{4}, xyz(x^{2}−y^{2}), xyz^{3}, x^{2}(x^{2}−3y^{2})^{2}−y^{2}(3x^{2}−y^{2})^{2}, z^{2}((x^{2}−y^{2})^{2}−4x^{2}y^{2}), z^{4}(x^{2}−y^{2}), z^{6} B1 R+p+d+2f+2g+3h+3i+4j+4k+5l+5m R_{z}, z, xy, z(x^{2}−y^{2}), z^{3}, xy(x^{2}−y^{2}), xyz^{2}, z((x^{2}−y^{2})^{2}−4x^{2}y^{2}), z^{3}(x^{2}−y^{2}), z^{5}, xy(x^{2}−3y^{2})(3x^{2}−y^{2}), xyz^{2}(x^{2}−y^{2}), xyz^{4} B2 R+p+d+2f+2g+3h+3i+4j+4k+5l+5m R_{y}, y, xz, y(3x^{2}−y^{2}), yz^{2}, xz(x^{2}−3y^{2}), xz^{3}, y((5+2√5)x^{2}−y^{2})((5−2√5)x^{2}−y^{2}), yz^{2}(3x^{2}−y^{2}), yz^{4}, xz(x^{2}−(5+2√5)y^{2})(x^{2}−(5−2√5)y^{2}), xz^{3}(x^{2}−3y^{2}), xz^{5} B3 R+p+d+2f+2g+3h+3i+4j+4k+5l+5m R_{x}, x, yz, x(x^{2}−3y^{2}), xz^{2}, yz(3x^{2}−y^{2}), yz^{3}, x(x^{2}−(5+2√5)y^{2})(x^{2}−(5−2√5)y^{2}), xz^{2}(x^{2}−3y^{2}), xz^{4}, yz((5+2√5)x^{2}−y^{2})((5−2√5)x^{2}−y^{2}), yz^{3}(3x^{2}−y^{2}), yz^{5} Notes: α The order of the D_{2} point group is 4, and the order of the principal axis (C_{2}) is 2. The group has 4 irreducible representations. β The D_{2} point group is also known as V. The letter V derives from German ‘Vierergruppe’ (group of four) for the Klein four-group. γ The D_{2} point group is isomorphic to C_{2v} and C_{2h}, and also to the Klein four-group. δ The D_{2} point group is generated by any two perpendicular C_{2} axes, canonically C_{2} and C_{2}^{′}. ε The D_{2} group has three nonequivalent C_{2} axes. The one labelled “C_{2}” is the principal axis (z in canonical orientation). The other two are perpendicular to the principal axis. By convention, the C_{2}^{′} is the x axis and C_{2}^{″} the y axis. ζ The lowest nonvanishing multipole moment in D_{2} is 4 (quadrupole moment). η This is an Abelian point group (the commutative law holds between all symmetry operations). The D_{2} group is Abelian because it satisfies the sufficient condition to contain no axes of order higher than two. In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional. θ The point group is chiral, as it does not contain any mirroring operation. ι The naming of irreducible representations in this group as B_{1},B_{2},B_{3} is purely conventional. κ 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. λ 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_{1} | ||
C_{2} C_{2v} C_{2h} | D_{2} | D_{2h} D_{2d} C_{i} |
D_{3} | ||
D_{4} |
This Character Table for the D_{2} point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.