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E64 2.00000 -1.99094 1.96386 -1.91899 1.85674 -1.77767 1.68251 -1.57211 1.44747 -1.30972 1.16011 -1.00000 0.83083 -0.65414 0.47152 -0.28463 0.09516 0.09516 -0.28463 0.47152 -0.65414 0.83083 -1.00000 1.16011 -1.30972 1.44747 -1.57211 1.68251 -1.77767 1.85674 -1.91899 1.96386 -1.99094 2.00000 -1.99094 1.96386 -1.91899 1.85674 -1.77767 1.68251 -1.57211 1.44747 -1.30972 1.16011 -1.00000 0.83083 -0.65414 0.47152 -0.28463 0.09516 0.09516 -0.28463 0.47152 -0.65414 0.83083 -1.00000 1.16011 -1.30972 1.44747 -1.57211 1.68251 -1.77767 1.85674 -1.91899 1.96386 -1.99094 2.00000 0.00000 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E65 2.00000 -1.99773 1.99094 -1.97964 1.96386 -1.94362 1.91899 -1.89000 1.85674 -1.81926 1.77767 -1.73205 1.68251 -1.62915 1.57211 -1.51150 1.44747 -1.38016 1.30972 -1.23632 1.16011 -1.08128 1.00000 -0.91645 0.83083 -0.74332 0.65414 -0.56347 0.47152 -0.37850 0.28463 -0.19011 0.09516 0.00000 -0.09516 0.19011 -0.28463 0.37850 -0.47152 0.56347 -0.65414 0.74332 -0.83083 0.91645 -1.00000 1.08128 -1.16011 1.23632 -1.30972 1.38016 -1.44747 1.51150 -1.57211 1.62915 -1.68251 1.73205 -1.77767 1.81926 -1.85674 1.89000 -1.91899 1.94362 -1.96386 1.97964 -1.99094 1.99773 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.997734678366 = 2*cos(2*π/132) = 2*cos(π/66) 1.990943845146 = 2*cos(4*π/132) = 2*cos(π/33) 1.979642883762 = 2*cos(6*π/132) = 2*cos(π/22) 1.963857394525 = 2*cos(8*π/132) = 2*cos(2*π/33) 1.943623136647 = 2*cos(10*π/132) = 2*cos(5*π/66) 1.918985947229 = 2*cos(12*π/132) = 2*cos(π/11) 1.890001637429 = 2*cos(14*π/132) = 2*cos(7*π/66) 1.856735866032 = 2*cos(16*π/132) = 2*cos(4*π/33) 1.819263990709 = 2*cos(18*π/132) = 2*cos(3*π/22) 1.777670897310 = 2*cos(20*π/132) = 2*cos(5*π/33) 1.732050807569 = 2*cos(22*π/132) = 2*cos(π/6) = √3 1.682507065662 = 2*cos(24*π/132) = 2*cos(2*π/11) 1.629151904101 = 2*cos(26*π/132) = 2*cos(13*π/66) 1.572106189486 = 2*cos(28*π/132) = 2*cos(7*π/33) 1.511499148709 = 2*cos(30*π/132) = 2*cos(5*π/22) 1.447468076210 = 2*cos(32*π/132) = 2*cos(8*π/33) 1.380158022964 = 2*cos(34*π/132) = 2*cos(17*π/66) 1.309721467891 = 2*cos(36*π/132) = 2*cos(3*π/11) 1.236317972441 = 2*cos(38*π/132) = 2*cos(19*π/66) 1.160113819142 = 2*cos(40*π/132) = 2*cos(10*π/33) 1.081281634911 = 2*cos(42*π/132) = 2*cos(7*π/22) 0.916453043455 = 2*cos(46*π/132) = 2*cos(23*π/66) 0.830830026004 = 2*cos(48*π/132) = 2*cos(4*π/11) 0.743324911321 = 2*cos(50*π/132) = 2*cos(25*π/66) 0.654135926635 = 2*cos(52*π/132) = 2*cos(13*π/33) 0.563465113683 = 2*cos(54*π/132) = 2*cos(9*π/22) 0.471517871019 = 2*cos(56*π/132) = 2*cos(14*π/33) 0.378502488721 = 2*cos(58*π/132) = 2*cos(29*π/66) 0.284629676547 = 2*cos(60*π/132) = 2*cos(5*π/11) 0.190112086608 = 2*cos(62*π/132) = 2*cos(31*π/66) 0.095163831647 = 2*cos(64*π/132) = 2*cos(16*π/33) 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 E57 m E58 l E59 k+m E60 j+l E61 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)} E62 h+j+l {z((x2−y2)2−4x2y2), xyz(x2−y2)} E63 g+i+k+m {xz(x2−3y2), yz(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E64 f+h+j+l {z(x2−y2), xyz}, {z3(x2−y2), xyz3} E65 R+d+g+i+k+m {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} Notes: α The order of the D66d point group is 264, and the order of the principal axis (S132) is 132. The group has 69 irreducible representations. β The D66d point group is isomorphic to C132v and D132. γ The D66d point group is generated by two symmetry elements, S132 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 D66d 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. For this group, however, some of the irrational characters cannot be expressed by a closed algebraic form using real numbers only.
D64d | ||
D65d | ||
C66 C66v C66h D66 D66h | D66d | S66 |
D67d | ||
D68d |
This Character Table for the D66d point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.