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E31 2.00000 -1.36035 -0.14946 1.56366 -1.97766 1.12664 0.44504 -1.73205 1.91115 -0.86777 -0.73068 1.86175 -1.80194 0.58951 1.00000 -1.94986 1.65248 -0.29808 -1.24698 1.99441 -1.46610 0.00000 1.46610 -1.99441 1.24698 0.29808 -1.65248 1.94986 -1.00000 -0.58951 1.80194 -1.86175 0.73068 0.86777 -1.91115 1.73205 -0.44504 -1.12664 1.97766 -1.56366 0.14946 1.36035 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E32 2.00000 -1.46610 0.14946 1.24698 -1.97766 1.65248 -0.44504 -1.00000 1.91115 -1.80194 0.73068 0.73068 -1.80194 1.91115 -1.00000 -0.44504 1.65248 -1.97766 1.24698 0.14946 -1.46610 2.00000 -1.46610 0.14946 1.24698 -1.97766 1.65248 -0.44504 -1.00000 1.91115 -1.80194 0.73068 0.73068 -1.80194 1.91115 -1.00000 -0.44504 1.65248 -1.97766 1.24698 0.14946 -1.46610 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E33 2.00000 -1.56366 0.44504 0.86777 -1.80194 1.94986 -1.24698 0.00000 1.24698 -1.94986 1.80194 -0.86777 -0.44504 1.56366 -2.00000 1.56366 -0.44504 -0.86777 1.80194 -1.94986 1.24698 0.00000 -1.24698 1.94986 -1.80194 0.86777 0.44504 -1.56366 2.00000 -1.56366 0.44504 0.86777 -1.80194 1.94986 -1.24698 0.00000 1.24698 -1.94986 1.80194 -0.86777 -0.44504 1.56366 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E34 2.00000 -1.65248 0.73068 0.44504 -1.46610 1.97766 -1.80194 1.00000 0.14946 -1.24698 1.91115 -1.91115 1.24698 -0.14946 -1.00000 1.80194 -1.97766 1.46610 -0.44504 -0.73068 1.65248 -2.00000 1.65248 -0.73068 -0.44504 1.46610 -1.97766 1.80194 -1.00000 -0.14946 1.24698 -1.91115 1.91115 -1.24698 0.14946 1.00000 -1.80194 1.97766 -1.46610 0.44504 0.73068 -1.65248 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E35 2.00000 -1.73205 1.00000 0.00000 -1.00000 1.73205 -2.00000 1.73205 -1.00000 0.00000 1.00000 -1.73205 2.00000 -1.73205 1.00000 0.00000 -1.00000 1.73205 -2.00000 1.73205 -1.00000 0.00000 1.00000 -1.73205 2.00000 -1.73205 1.00000 0.00000 -1.00000 1.73205 -2.00000 1.73205 -1.00000 0.00000 1.00000 -1.73205 2.00000 -1.73205 1.00000 0.00000 -1.00000 1.73205 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E36 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 -1.80194 1.24698 -0.44504 -0.44504 1.24698 -1.80194 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E37 2.00000 -1.86175 1.46610 -0.86777 0.14946 0.58951 -1.24698 1.73205 -1.97766 1.94986 -1.65248 1.12664 -0.44504 -0.29808 1.00000 -1.56366 1.91115 -1.99441 1.80194 -1.36035 0.73068 0.00000 -0.73068 1.36035 -1.80194 1.99441 -1.91115 1.56366 -1.00000 0.29808 0.44504 -1.12664 1.65248 -1.94986 1.97766 -1.73205 1.24698 -0.58951 -0.14946 0.86777 -1.46610 1.86175 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ..TT......... E38 2.00000 -1.91115 1.65248 -1.24698 0.73068 -0.14946 -0.44504 1.00000 -1.46610 1.80194 -1.97766 1.97766 -1.80194 1.46610 -1.00000 0.44504 0.14946 -0.73068 1.24698 -1.65248 1.91115 -2.00000 1.91115 -1.65248 1.24698 -0.73068 0.14946 0.44504 -1.00000 1.46610 -1.80194 1.97766 -1.97766 1.80194 -1.46610 1.00000 -0.44504 -0.14946 0.73068 -1.24698 1.65248 -1.91115 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ..TT....... ............. E39 2.00000 -1.94986 1.80194 -1.56366 1.24698 -0.86777 0.44504 0.00000 -0.44504 0.86777 -1.24698 1.56366 -1.80194 1.94986 -2.00000 1.94986 -1.80194 1.56366 -1.24698 0.86777 -0.44504 0.00000 0.44504 -0.86777 1.24698 -1.56366 1.80194 -1.94986 2.00000 -1.94986 1.80194 -1.56366 1.24698 -0.86777 0.44504 0.00000 -0.44504 0.86777 -1.24698 1.56366 -1.80194 1.94986 -2.00000 0.00000 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E40 2.00000 -1.97766 1.91115 -1.80194 1.65248 -1.46610 1.24698 -1.00000 0.73068 -0.44504 0.14946 0.14946 -0.44504 0.73068 -1.00000 1.24698 -1.46610 1.65248 -1.80194 1.91115 -1.97766 2.00000 -1.97766 1.91115 -1.80194 1.65248 -1.46610 1.24698 -1.00000 0.73068 -0.44504 0.14946 0.14946 -0.44504 0.73068 -1.00000 1.24698 -1.46610 1.65248 -1.80194 1.91115 -1.97766 2.00000 0.00000 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E41 2.00000 -1.99441 1.97766 -1.94986 1.91115 -1.86175 1.80194 -1.73205 1.65248 -1.56366 1.46610 -1.36035 1.24698 -1.12664 1.00000 -0.86777 0.73068 -0.58951 0.44504 -0.29808 0.14946 0.00000 -0.14946 0.29808 -0.44504 0.58951 -0.73068 0.86777 -1.00000 1.12664 -1.24698 1.36035 -1.46610 1.56366 -1.65248 1.73205 -1.80194 1.86175 -1.91115 1.94986 -1.97766 1.99441 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.994407594362 = 2*cos(2*π/84) = 2*cos(π/42) 1.977661652450 = 2*cos(4*π/84) = 2*cos(π/21) 1.949855824364 = 2*cos(6*π/84) = 2*cos(π/14) 1.911145611572 = 2*cos(8*π/84) = 2*cos(2*π/21) 1.861747497288 = 2*cos(10*π/84) = 2*cos(5*π/42) 1.801937735805 = 2*cos(12*π/84) = 2*cos(π/7) 1.732050807569 = 2*cos(14*π/84) = 2*cos(π/6) = √3 1.652477548632 = 2*cos(16*π/84) = 2*cos(4*π/21) 1.563662964936 = 2*cos(18*π/84) = 2*cos(3*π/14) 1.466103743660 = 2*cos(20*π/84) = 2*cos(5*π/21) 1.360345475542 = 2*cos(22*π/84) = 2*cos(11*π/42) 1.246979603717 = 2*cos(24*π/84) = 2*cos(2*π/7) 1.126640116127 = 2*cos(26*π/84) = 2*cos(13*π/42) 0.867767478235 = 2*cos(30*π/84) = 2*cos(5*π/14) 0.730682048733 = 2*cos(32*π/84) = 2*cos(8*π/21) 0.589510348822 = 2*cos(34*π/84) = 2*cos(17*π/42) 0.445041867913 = 2*cos(36*π/84) = 2*cos(3*π/7) 0.298084532352 = 2*cos(38*π/84) = 2*cos(19*π/42) 0.149460187173 = 2*cos(40*π/84) = 2*cos(10*π/21) Symmetry of Rotations and Cartesian products A1 d+g+i+k+m z2, z4, z6 A2 R Rz B2 p+f+h+j+l z, z3, z5 E1 p+f+h+j+l {x, y}, {xz2, yz2}, {xz4, yz4} E2 d+g+i+k+m {x2−y2, xy}, {z2(x2−y2), xyz2}, {z4(x2−y2), xyz4} E3 f+h+j+l {x(x2−3y2), y(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)} E4 g+i+k+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E5 h+j+l {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)} E6 i+k+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E7 j+l E8 k+m E9 l E10 m E33 m E34 l E35 k+m E36 j+l E37 i+k+m {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)} E38 h+j+l {z((x2−y2)2−4x2y2), xyz(x2−y2)} E39 g+i+k+m {xz(x2−3y2), yz(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E40 f+h+j+l {z(x2−y2), xyz}, {z3(x2−y2), xyz3} E41 R+d+g+i+k+m {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} Notes: α The order of the D42d point group is 168, and the order of the principal axis (S84) is 84. The group has 45 irreducible representations. β The D42d point group is isomorphic to C84v and D84. γ The D42d point group is generated by two symmetry elements, S84 and either a perpendicular C2′ or a vertical σd. Also, the group may be generated from a C2′ plus a σd (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ The group contains one set of twofold symmetry axes (C2′) perpendicular to the principal (z) axis. Both x and y axes are members of that set. ε The single σd set of symmetry planes contains neither the xz nor the yz planes; but it contains the median plane (x+y)z. ζ The lowest nonvanishing multipole moment in D42d is 4 (quadrupole moment). η This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. θ Some of the characters in the table are irrational because the order of the principal axis is neither 1,2,3,4 nor 6. These irrational values can be expressed as cosine values, or as solutions of algebraic equations with a leading coefficient of 1. All characters are algebraic integers of a degree much less than half the order of the principal axis. ι The point group corresponds to a polygon inconstructible by the classical means of ruler and compass. Yet it becomes constructible if angle trisection is allowed, e.g., with neusis construction or origami. This is because the order of the principal axis is given by a product of any number of different Pierpont primes (...,5,7,13,17,19,37,73,97,109,163,...) times arbitrary powers of two and three. While some characters of this group can be expressed using real integers and square roots alone, others need complex numbers and third roots.
D40d | ||
D41d | ||
C42 C42v C42h D42 D42h | D42d | S42 |
D43d | ||
D44d |
This Character Table for the D42d point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.