D53 E 2 C53 2 C53^2 2 C53^3 2 C53^4 2 C53^5 2 C53^6 2 C53^7 2 C53^8 2 C53^9 2 C53^10 2 C53^11 2 C53^12 2 C53^13 2 C53^14 2 C53^15 2 C53^16 2 C53^17 2 C53^18 2 C53^19 2 C53^20 2 C53^21 2 C53^22 2 C53^23 2 C53^24 2 C53^25 2 C53^26 53 C2' <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 ... ... ....T ....... ........T ........... ............T A2 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 ..T ..T ..... ......T ......... ..........T ............. E1 2.00000 1.98596 1.94405 1.87484 1.77931 1.65881 1.51502 1.34997 1.16596 0.96558 0.75166 0.52717 0.29529 0.05927 -0.17759 -0.41196 -0.64054 -0.86013 -1.06765 -1.26018 -1.43501 -1.58971 -1.72209 -1.83029 -1.91280 -1.96846 -1.99649 0.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.00000 1.94405 1.77931 1.51502 1.16596 0.75166 0.29529 -0.17759 -0.64054 -1.06765 -1.43501 -1.72209 -1.91280 -1.99649 -1.96846 -1.83029 -1.58971 -1.26018 -0.86013 -0.41196 0.05927 0.52717 0.96558 1.34997 1.65881 1.87484 1.98596 0.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.00000 1.87484 1.51502 0.96558 0.29529 -0.41196 -1.06765 -1.58971 -1.91280 -1.99649 -1.83029 -1.43501 -0.86013 -0.17759 0.52717 1.16596 1.65881 1.94405 1.98596 1.77931 1.34997 0.75166 0.05927 -0.64054 -1.26018 -1.72209 -1.96846 0.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 2.00000 1.77931 1.16596 0.29529 -0.64054 -1.43501 -1.91280 -1.96846 -1.58971 -0.86013 0.05927 0.96558 1.65881 1.98596 1.87484 1.34997 0.52717 -0.41196 -1.26018 -1.83029 -1.99649 -1.72209 -1.06765 -0.17759 0.75166 1.51502 1.94405 0.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 2.00000 1.65881 0.75166 -0.41196 -1.43501 -1.96846 -1.83029 -1.06765 0.05927 1.16596 1.87484 1.94405 1.34997 0.29529 -0.86013 -1.72209 -1.99649 -1.58971 -0.64054 0.52717 1.51502 1.98596 1.77931 0.96558 -0.17759 -1.26018 -1.91280 0.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 2.00000 1.51502 0.29529 -1.06765 -1.91280 -1.83029 -0.86013 0.52717 1.65881 1.98596 1.34997 0.05927 -1.26018 -1.96846 -1.72209 -0.64054 0.75166 1.77931 1.94405 1.16596 -0.17759 -1.43501 -1.99649 -1.58971 -0.41196 0.96558 1.87484 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.34997 -0.17759 -1.58971 -1.96846 -1.06765 0.52717 1.77931 1.87484 0.75166 -0.86013 -1.91280 -1.72209 -0.41196 1.16596 1.98596 1.51502 0.05927 -1.43501 -1.99649 -1.26018 0.29529 1.65881 1.94405 0.96558 -0.64054 -1.83029 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.16596 -0.64054 -1.91280 -1.58971 0.05927 1.65881 1.87484 0.52717 -1.26018 -1.99649 -1.06765 0.75166 1.94405 1.51502 -0.17759 -1.72209 -1.83029 -0.41196 1.34997 1.98596 0.96558 -0.86013 -1.96846 -1.43501 0.29529 1.77931 0.00000 ... ... ..... ....... ......... ........... ............. E9 2.00000 0.96558 -1.06765 -1.99649 -0.86013 1.16596 1.98596 0.75166 -1.26018 -1.96846 -0.64054 1.34997 1.94405 0.52717 -1.43501 -1.91280 -0.41196 1.51502 1.87484 0.29529 -1.58971 -1.83029 -0.17759 1.65881 1.77931 0.05927 -1.72209 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 0.75166 -1.43501 -1.83029 0.05927 1.87484 1.34997 -0.86013 -1.99649 -0.64054 1.51502 1.77931 -0.17759 -1.91280 -1.26018 0.96558 1.98596 0.52717 -1.58971 -1.72209 0.29529 1.94405 1.16596 -1.06765 -1.96846 -0.41196 1.65881 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 0.52717 -1.72209 -1.43501 0.96558 1.94405 0.05927 -1.91280 -1.06765 1.34997 1.77931 -0.41196 -1.99649 -0.64054 1.65881 1.51502 -0.86013 -1.96846 -0.17759 1.87484 1.16596 -1.26018 -1.83029 0.29529 1.98596 0.75166 -1.58971 0.00000 ... ... ..... ....... ......... ........... ............. E12 2.00000 0.29529 -1.91280 -0.86013 1.65881 1.34997 -1.26018 -1.72209 0.75166 1.94405 -0.17759 -1.99649 -0.41196 1.87484 0.96558 -1.58971 -1.43501 1.16596 1.77931 -0.64054 -1.96846 0.05927 1.98596 0.52717 -1.83029 -1.06765 1.51502 0.00000 ... ... ..... ....... ......... ........... ............. E13 2.00000 0.05927 -1.99649 -0.17759 1.98596 0.29529 -1.96846 -0.41196 1.94405 0.52717 -1.91280 -0.64054 1.87484 0.75166 -1.83029 -0.86013 1.77931 0.96558 -1.72209 -1.06765 1.65881 1.16596 -1.58971 -1.26018 1.51502 1.34997 -1.43501 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 -0.17759 -1.96846 0.52717 1.87484 -0.86013 -1.72209 1.16596 1.51502 -1.43501 -1.26018 1.65881 0.96558 -1.83029 -0.64054 1.94405 0.29529 -1.99649 0.05927 1.98596 -0.41196 -1.91280 0.75166 1.77931 -1.06765 -1.58971 1.34997 0.00000 ... ... ..... ....... ......... ........... ............. E15 2.00000 -0.41196 -1.83029 1.16596 1.34997 -1.72209 -0.64054 1.98596 -0.17759 -1.91280 0.96558 1.51502 -1.58971 -0.86013 1.94405 0.05927 -1.96846 0.75166 1.65881 -1.43501 -1.06765 1.87484 0.29529 -1.99649 0.52717 1.77931 -1.26018 0.00000 ... ... ..... ....... ......... ........... ............. E16 2.00000 -0.64054 -1.58971 1.65881 0.52717 -1.99649 0.75166 1.51502 -1.72209 -0.41196 1.98596 -0.86013 -1.43501 1.77931 0.29529 -1.96846 0.96558 1.34997 -1.83029 -0.17759 1.94405 -1.06765 -1.26018 1.87484 0.05927 -1.91280 1.16596 0.00000 ... ... ..... ....... ......... ........... ............. E17 2.00000 -0.86013 -1.26018 1.94405 -0.41196 -1.58971 1.77931 0.05927 -1.83029 1.51502 0.52717 -1.96846 1.16596 0.96558 -1.99649 0.75166 1.34997 -1.91280 0.29529 1.65881 -1.72209 -0.17759 1.87484 -1.43501 -0.64054 1.98596 -1.06765 0.00000 ... ... ..... ....... ......... ........... ............. E18 2.00000 -1.06765 -0.86013 1.98596 -1.26018 -0.64054 1.94405 -1.43501 -0.41196 1.87484 -1.58971 -0.17759 1.77931 -1.72209 0.05927 1.65881 -1.83029 0.29529 1.51502 -1.91280 0.52717 1.34997 -1.96846 0.75166 1.16596 -1.99649 0.96558 0.00000 ... ... ..... ....... ......... ........... ............. E19 2.00000 -1.26018 -0.41196 1.77931 -1.83029 0.52717 1.16596 -1.99649 1.34997 0.29529 -1.72209 1.87484 -0.64054 -1.06765 1.98596 -1.43501 -0.17759 1.65881 -1.91280 0.75166 0.96558 -1.96846 1.51502 0.05927 -1.58971 1.94405 -0.86013 0.00000 ... ... ..... ....... ......... ........... ............. E20 2.00000 -1.43501 0.05927 1.34997 -1.99649 1.51502 -0.17759 -1.26018 1.98596 -1.58971 0.29529 1.16596 -1.96846 1.65881 -0.41196 -1.06765 1.94405 -1.72209 0.52717 0.96558 -1.91280 1.77931 -0.64054 -0.86013 1.87484 -1.83029 0.75166 0.00000 ... ... ..... ....... ......... ........... ............. E21 2.00000 -1.58971 0.52717 0.75166 -1.72209 1.98596 -1.43501 0.29529 0.96558 -1.83029 1.94405 -1.26018 0.05927 1.16596 -1.91280 1.87484 -1.06765 -0.17759 1.34997 -1.96846 1.77931 -0.86013 -0.41196 1.51502 -1.99649 1.65881 -0.64054 0.00000 ... ... ..... ....... ......... ........... ............. E22 2.00000 -1.72209 0.96558 0.05927 -1.06765 1.77931 -1.99649 1.65881 -0.86013 -0.17759 1.16596 -1.83029 1.98596 -1.58971 0.75166 0.29529 -1.26018 1.87484 -1.96846 1.51502 -0.64054 -0.41196 1.34997 -1.91280 1.94405 -1.43501 0.52717 0.00000 ... ... ..... ....... ......... ........... ............. E23 2.00000 -1.83029 1.34997 -0.64054 -0.17759 0.96558 -1.58971 1.94405 -1.96846 1.65881 -1.06765 0.29529 0.52717 -1.26018 1.77931 -1.99649 1.87484 -1.43501 0.75166 0.05927 -0.86013 1.51502 -1.91280 1.98596 -1.72209 1.16596 -0.41196 0.00000 ... ... ..... ....... ......... ........... ............. E24 2.00000 -1.91280 1.65881 -1.26018 0.75166 -0.17759 -0.41196 0.96558 -1.43501 1.77931 -1.96846 1.98596 -1.83029 1.51502 -1.06765 0.52717 0.05927 -0.64054 1.16596 -1.58971 1.87484 -1.99649 1.94405 -1.72209 1.34997 -0.86013 0.29529 0.00000 ... ... ..... ....... ......... ........... ............. E25 2.00000 -1.96846 1.87484 -1.72209 1.51502 -1.26018 0.96558 -0.64054 0.29529 0.05927 -0.41196 0.75166 -1.06765 1.34997 -1.58971 1.77931 -1.91280 1.98596 -1.99649 1.94405 -1.83029 1.65881 -1.43501 1.16596 -0.86013 0.52717 -0.17759 0.00000 ... ... ..... ....... ......... ........... ............. E26 2.00000 -1.99649 1.98596 -1.96846 1.94405 -1.91280 1.87484 -1.83029 1.77931 -1.72209 1.65881 -1.58971 1.51502 -1.43501 1.34997 -1.26018 1.16596 -1.06765 0.96558 -0.86013 0.75166 -0.64054 0.52717 -0.41196 0.29529 -0.17759 0.05927 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.985962192027 = 2*cos(2*π/53) 1.944045828161 = 2*cos(4*π/53) 1.874839322268 = 2*cos(6*π/53) 1.779314181989 = 2*cos(8*π/53) 1.658811370900 = 2*cos(10*π/53) 1.515022484323 = 2*cos(12*π/53) 1.349966003036 = 2*cos(14*π/53) 1.165958958229 = 2*cos(16*π/53) 0.965584405461 = 2*cos(18*π/53) 0.751655164228 = 2*cos(20*π/53) 0.527174332138 = 2*cos(22*π/53) 0.295293128005 = 2*cos(24*π/53) 0.059266655645 = 2*cos(26*π/53) -0.177591790646 = 2*cos(28*π/53) -0.411957237482 = 2*cos(30*π/53) -0.640539707726 = 2*cos(32*π/53) -0.860130404553 = 2*cos(34*π/53) -1.067646755930 = 2*cos(36*π/53) -1.260175687163 = 2*cos(38*π/53) -1.435014514089 = 2*cos(40*π/53) -1.589708882827 = 2*cos(42*π/53) -1.722087223535 = 2*cos(44*π/53) -1.830291234486 = 2*cos(46*π/53) -1.912801968553 = 2*cos(48*π/53) -1.968461155895 = 2*cos(50*π/53) -1.996487463529 = 2*cos(52*π/53) 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 D53 point group is 106, and the order of the principal axis (C53) is 53. The group has 28 irreducible representations. β The D53 point group is isomorphic to C53v. γ The D53 point group is generated by two symmetry elements, C53 and a perpendicular C2′. Also, the group may be generated from any two C2′ axes. δ The group contains one set of C2′ symmetry axes perpendicular to the principal (z) axis. The x axis (but not the y axis) is a member of that set. ε The lowest nonvanishing multipole moment in D53 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 just less than half the order of the principal axis. For this group, however, none of the irrational characters can be expressed by a closed algebraic form using real numbers only.
| D51 | ||
| D52 | ||
| C53 C53v C53h | D53 | D53h D53d |
| D54 | ||
| D55 |
This Character Table for the D53 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.