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E27 2.00000 -0.69893 -1.51150 1.75536 0.28463 -1.95429 1.08128 1.19856 -1.91899 0.14268 1.81926 -1.41421 -0.83083 1.99490 -0.56347 -1.60108 1.68251 0.42513 -1.97964 0.95850 1.30972 -1.87390 0.00000 1.87390 -1.30972 -0.95850 1.97964 -0.42513 -1.68251 1.60108 0.56347 -1.99490 0.83083 1.41421 -1.81926 -0.14268 1.91899 -1.19856 -1.08128 1.95429 -0.28463 -1.75536 1.51150 0.69893 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E28 2.00000 -0.83083 -1.30972 1.91899 -0.28463 -1.68251 1.68251 0.28463 -1.91899 1.30972 0.83083 -2.00000 0.83083 1.30972 -1.91899 0.28463 1.68251 -1.68251 -0.28463 1.91899 -1.30972 -0.83083 2.00000 -0.83083 -1.30972 1.91899 -0.28463 -1.68251 1.68251 0.28463 -1.91899 1.30972 0.83083 -2.00000 0.83083 1.30972 -1.91899 0.28463 1.68251 -1.68251 -0.28463 1.91899 -1.30972 -0.83083 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E29 2.00000 -0.95850 -1.08128 1.99490 -0.83083 -1.19856 1.97964 -0.69893 -1.30972 1.95429 -0.56347 -1.41421 1.91899 -0.42513 -1.51150 1.87390 -0.28463 -1.60108 1.81926 -0.14268 -1.68251 1.75536 0.00000 -1.75536 1.68251 0.14268 -1.81926 1.60108 0.28463 -1.87390 1.51150 0.42513 -1.91899 1.41421 0.56347 -1.95429 1.30972 0.69893 -1.97964 1.19856 0.83083 -1.99490 1.08128 0.95850 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E30 2.00000 -1.08128 -0.83083 1.97964 -1.30972 -0.56347 1.91899 -1.51150 -0.28463 1.81926 -1.68251 0.00000 1.68251 -1.81926 0.28463 1.51150 -1.91899 0.56347 1.30972 -1.97964 0.83083 1.08128 -2.00000 1.08128 0.83083 -1.97964 1.30972 0.56347 -1.91899 1.51150 0.28463 -1.81926 1.68251 0.00000 -1.68251 1.81926 -0.28463 -1.51150 1.91899 -0.56347 -1.30972 1.97964 -0.83083 -1.08128 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E31 2.00000 -1.19856 -0.56347 1.87390 -1.68251 0.14268 1.51150 -1.95429 0.83083 0.95850 -1.97964 1.41421 0.28463 -1.75536 1.81926 -0.42513 -1.30972 1.99490 -1.08128 -0.69893 1.91899 -1.60108 0.00000 1.60108 -1.91899 0.69893 1.08128 -1.99490 1.30972 0.42513 -1.81926 1.75536 -0.28463 -1.41421 1.97964 -0.95850 -0.83083 1.95429 -1.51150 -0.14268 1.68251 -1.87390 0.56347 1.19856 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E32 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E33 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E34 2.00000 -1.51150 0.28463 1.08128 -1.91899 1.81926 -0.83083 -0.56347 1.68251 -1.97964 1.30972 0.00000 -1.30972 1.97964 -1.68251 0.56347 0.83083 -1.81926 1.91899 -1.08128 -0.28463 1.51150 -2.00000 1.51150 -0.28463 -1.08128 1.91899 -1.81926 0.83083 0.56347 -1.68251 1.97964 -1.30972 0.00000 1.30972 -1.97964 1.68251 -0.56347 -0.83083 1.81926 -1.91899 1.08128 0.28463 -1.51150 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E35 2.00000 -1.60108 0.56347 0.69893 -1.68251 1.99490 -1.51150 0.42513 0.83083 -1.75536 1.97964 -1.41421 0.28463 0.95850 -1.81926 1.95429 -1.30972 0.14268 1.08128 -1.87390 1.91899 -1.19856 0.00000 1.19856 -1.91899 1.87390 -1.08128 -0.14268 1.30972 -1.95429 1.81926 -0.95850 -0.28463 1.41421 -1.97964 1.75536 -0.83083 -0.42513 1.51150 -1.99490 1.68251 -0.69893 -0.56347 1.60108 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E36 2.00000 -1.68251 0.83083 0.28463 -1.30972 1.91899 -1.91899 1.30972 -0.28463 -0.83083 1.68251 -2.00000 1.68251 -0.83083 -0.28463 1.30972 -1.91899 1.91899 -1.30972 0.28463 0.83083 -1.68251 2.00000 -1.68251 0.83083 0.28463 -1.30972 1.91899 -1.91899 1.30972 -0.28463 -0.83083 1.68251 -2.00000 1.68251 -0.83083 -0.28463 1.30972 -1.91899 1.91899 -1.30972 0.28463 0.83083 -1.68251 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E37 2.00000 -1.75536 1.08128 -0.14268 -0.83083 1.60108 -1.97964 1.87390 -1.30972 0.42513 0.56347 -1.41421 1.91899 -1.95429 1.51150 -0.69893 -0.28463 1.19856 -1.81926 1.99490 -1.68251 0.95850 0.00000 -0.95850 1.68251 -1.99490 1.81926 -1.19856 0.28463 0.69893 -1.51150 1.95429 -1.91899 1.41421 -0.56347 -0.42513 1.30972 -1.87390 1.97964 -1.60108 0.83083 0.14268 -1.08128 1.75536 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E38 2.00000 -1.81926 1.30972 -0.56347 -0.28463 1.08128 -1.68251 1.97964 -1.91899 1.51150 -0.83083 0.00000 0.83083 -1.51150 1.91899 -1.97964 1.68251 -1.08128 0.28463 0.56347 -1.30972 1.81926 -2.00000 1.81926 -1.30972 0.56347 0.28463 -1.08128 1.68251 -1.97964 1.91899 -1.51150 0.83083 0.00000 -0.83083 1.51150 -1.91899 1.97964 -1.68251 1.08128 -0.28463 -0.56347 1.30972 -1.81926 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E39 2.00000 -1.87390 1.51150 -0.95850 0.28463 0.42513 -1.08128 1.60108 -1.91899 1.99490 -1.81926 1.41421 -0.83083 0.14268 0.56347 -1.19856 1.68251 -1.95429 1.97964 -1.75536 1.30972 -0.69893 0.00000 0.69893 -1.30972 1.75536 -1.97964 1.95429 -1.68251 1.19856 -0.56347 -0.14268 0.83083 -1.41421 1.81926 -1.99490 1.91899 -1.60108 1.08128 -0.42513 -0.28463 0.95850 -1.51150 1.87390 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E40 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E41 2.00000 -1.95429 1.81926 -1.60108 1.30972 -0.95850 0.56347 -0.14268 -0.28463 0.69893 -1.08128 1.41421 -1.68251 1.87390 -1.97964 1.99490 -1.91899 1.75536 -1.51150 1.19856 -0.83083 0.42513 0.00000 -0.42513 0.83083 -1.19856 1.51150 -1.75536 1.91899 -1.99490 1.97964 -1.87390 1.68251 -1.41421 1.08128 -0.69893 0.28463 0.14268 -0.56347 0.95850 -1.30972 1.60108 -1.81926 1.95429 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E42 2.00000 -1.97964 1.91899 -1.81926 1.68251 -1.51150 1.30972 -1.08128 0.83083 -0.56347 0.28463 0.00000 -0.28463 0.56347 -0.83083 1.08128 -1.30972 1.51150 -1.68251 1.81926 -1.91899 1.97964 -2.00000 1.97964 -1.91899 1.81926 -1.68251 1.51150 -1.30972 1.08128 -0.83083 0.56347 -0.28463 0.00000 0.28463 -0.56347 0.83083 -1.08128 1.30972 -1.51150 1.68251 -1.81926 1.91899 -1.97964 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E43 2.00000 -1.99490 1.97964 -1.95429 1.91899 -1.87390 1.81926 -1.75536 1.68251 -1.60108 1.51150 -1.41421 1.30972 -1.19856 1.08128 -0.95850 0.83083 -0.69893 0.56347 -0.42513 0.28463 -0.14268 0.00000 0.14268 -0.28463 0.42513 -0.56347 0.69893 -0.83083 0.95850 -1.08128 1.19856 -1.30972 1.41421 -1.51150 1.60108 -1.68251 1.75536 -1.81926 1.87390 -1.91899 1.95429 -1.97964 1.99490 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.994904229221 = 2*cos(2*π/88) = 2*cos(π/44) 1.979642883762 = 2*cos(4*π/88) = 2*cos(π/22) 1.954293731942 = 2*cos(6*π/88) = 2*cos(3*π/44) 1.918985947229 = 2*cos(8*π/88) = 2*cos(π/11) 1.873899450000 = 2*cos(10*π/88) = 2*cos(5*π/44) 1.819263990709 = 2*cos(12*π/88) = 2*cos(3*π/22) 1.755357979135 = 2*cos(14*π/88) = 2*cos(7*π/44) 1.682507065662 = 2*cos(16*π/88) = 2*cos(2*π/11) 1.601082481849 = 2*cos(18*π/88) = 2*cos(9*π/44) 1.511499148709 = 2*cos(20*π/88) = 2*cos(5*π/22) 1.414213562373 = 2*cos(22*π/88) = 2*cos(π/4) = √2 1.309721467891 = 2*cos(24*π/88) = 2*cos(3*π/11) 1.198555333023 = 2*cos(26*π/88) = 2*cos(13*π/44) 1.081281634911 = 2*cos(28*π/88) = 2*cos(7*π/22) 0.958497973440 = 2*cos(30*π/88) = 2*cos(15*π/44) 0.830830026004 = 2*cos(32*π/88) = 2*cos(4*π/11) 0.698928359198 = 2*cos(34*π/88) = 2*cos(17*π/44) 0.563465113683 = 2*cos(36*π/88) = 2*cos(9*π/22) 0.425130579106 = 2*cos(38*π/88) = 2*cos(19*π/44) 0.284629676547 = 2*cos(40*π/88) = 2*cos(5*π/11) 0.142678366398 = 2*cos(42*π/88) = 2*cos(21*π/44) Symmetry of Rotations and Cartesian products A1 d+g+i+k+m z2, z4, z6 A2 R+p+f+h+j+l Rz, z, z3, z5 E1 R+p+d+f+g+h+i+j+k+l+m {Rx, Ry}, {x, y}, {xz, yz}, {xz2, yz2}, {xz3, yz3}, {xz4, yz4}, {xz5, yz5} E2 d+f+g+h+i+j+k+l+m {x2−y2, xy}, {z(x2−y2), xyz}, {z2(x2−y2), xyz2}, {z3(x2−y2), xyz3}, {z4(x2−y2), xyz4} E3 f+g+h+i+j+k+l+m {x(x2−3y2), y(3x2−y2)}, {xz(x2−3y2), yz(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E4 g+h+i+j+k+l+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z((x2−y2)2−4x2y2), xyz(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E5 h+i+j+k+l+m {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)} E6 i+j+k+l+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E7 j+k+l+m E8 k+l+m E9 l+m E10 m Notes: α The order of the D88 point group is 176, and the order of the principal axis (C88) is 88. The group has 47 irreducible representations. β The D88 point group is isomorphic to D44d and C88v. γ The D88 point group is generated by two symmetry elements, C88 and a perpendicular C2′ (or, non-canonically, C2″). Also, the group may be generated from a C2′ plus a C2″ (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ There are two different sets of twofold symmetry axes perpendicular to the principal axis (z axis in standard orientation). By convention, the set denoted as C2′ contains both the x and y axes. ε The lowest nonvanishing multipole moment in D88 is 4 (quadrupole 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 The point group is chiral, as it does not contain any mirroring operation. 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. For this group, however, some of the irrational characters cannot be expressed by a closed algebraic form using real numbers only.
| D86 | ||
| D87 | ||
| C88 C88v C88h | D88 | D88h D88d S88 |
| D89 | ||
| D90 |
This Character Table for the D88 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.