C31v E 2 C31 2 C31^2 2 C31^3 2 C31^4 2 C31^5 2 C31^6 2 C31^7 2 C31^8 2 C31^9 2 C31^10 2 C31^11 2 C31^12 2 C31^13 2 C31^14 2 C31^15 31 sv <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 ... ..T ....T ......T ........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 ..T ... ..... ....... ......... ........... ............. E1 2.00000 1.95906 1.83792 1.64153 1.37793 1.05793 0.69461 0.30286 -0.10130 -0.50131 -0.88079 -1.22421 -1.51752 -1.74869 -1.90828 -1.98974 0.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.00000 1.83792 1.37793 0.69461 -0.10130 -0.88079 -1.51752 -1.90828 -1.98974 -1.74869 -1.22421 -0.50131 0.30286 1.05793 1.64153 1.95906 0.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.00000 1.64153 0.69461 -0.50131 -1.51752 -1.98974 -1.74869 -0.88079 0.30286 1.37793 1.95906 1.83792 1.05793 -0.10130 -1.22421 -1.90828 0.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 2.00000 1.37793 -0.10130 -1.51752 -1.98974 -1.22421 0.30286 1.64153 1.95906 1.05793 -0.50131 -1.74869 -1.90828 -0.88079 0.69461 1.83792 0.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 2.00000 1.05793 -0.88079 -1.98974 -1.22421 0.69461 1.95906 1.37793 -0.50131 -1.90828 -1.51752 0.30286 1.83792 1.64153 -0.10130 -1.74869 0.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 2.00000 0.69461 -1.51752 -1.74869 0.30286 1.95906 1.05793 -1.22421 -1.90828 -0.10130 1.83792 1.37793 -0.88079 -1.98974 -0.50131 1.64153 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 0.30286 -1.90828 -0.88079 1.64153 1.37793 -1.22421 -1.74869 0.69461 1.95906 -0.10130 -1.98974 -0.50131 1.83792 1.05793 -1.51752 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 -0.10130 -1.98974 0.30286 1.95906 -0.50131 -1.90828 0.69461 1.83792 -0.88079 -1.74869 1.05793 1.64153 -1.22421 -1.51752 1.37793 0.00000 ... ... ..... ....... ......... ........... ............. E9 2.00000 -0.50131 -1.74869 1.37793 1.05793 -1.90828 -0.10130 1.95906 -0.88079 -1.51752 1.64153 0.69461 -1.98974 0.30286 1.83792 -1.22421 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 -0.88079 -1.22421 1.95906 -0.50131 -1.51752 1.83792 -0.10130 -1.74869 1.64153 0.30286 -1.90828 1.37793 0.69461 -1.98974 1.05793 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 -1.22421 -0.50131 1.83792 -1.74869 0.30286 1.37793 -1.98974 1.05793 0.69461 -1.90828 1.64153 -0.10130 -1.51752 1.95906 -0.88079 0.00000 ... ... ..... ....... ......... ........... ............. E12 2.00000 -1.51752 0.30286 1.05793 -1.90828 1.83792 -0.88079 -0.50131 1.64153 -1.98974 1.37793 -0.10130 -1.22421 1.95906 -1.74869 0.69461 0.00000 ... ... ..... ....... ......... ........... ............. E13 2.00000 -1.74869 1.05793 -0.10130 -0.88079 1.64153 -1.98974 1.83792 -1.22421 0.30286 0.69461 -1.51752 1.95906 -1.90828 1.37793 -0.50131 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 -1.90828 1.64153 -1.22421 0.69461 -0.10130 -0.50131 1.05793 -1.51752 1.83792 -1.98974 1.95906 -1.74869 1.37793 -0.88079 0.30286 0.00000 ... ... ..... ....... ......... ........... ............. E15 2.00000 -1.98974 1.95906 -1.90828 1.83792 -1.74869 1.64153 -1.51752 1.37793 -1.22421 1.05793 -0.88079 0.69461 -0.50131 0.30286 -0.10130 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.959059882505 = 2*cos(2*π/31) 1.837915623240 = 2*cos(4*π/31) 1.641526882415 = 2*cos(6*π/31) 1.377933838151 = 2*cos(8*π/31) 1.057928020654 = 2*cos(10*π/31) 0.694610505690 = 2*cos(12*π/31) 0.302855555009 = 2*cos(14*π/31) -0.101298337677 = 2*cos(16*π/31) -0.501305064517 = 2*cos(18*π/31) -0.880788303115 = 2*cos(20*π/31) -1.224211965095 = 2*cos(22*π/31) -1.517516245386 = 2*cos(24*π/31) -1.748693232289 = 2*cos(26*π/31) -1.908278512800 = 2*cos(28*π/31) -1.989738646784 = 2*cos(30*π/31) Symmetry of Rotations and Cartesian products A1 p+d+f+g+h+i+j+k+l+m z, z2, z3, z4, z5, z6 A2 R Rz 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 C31v point group is 62, and the order of the principal axis (C31) is 31. The group has 17 irreducible representations. β The C31v point group is isomorphic to D31. γ The C31v point group is generated by two symmetry elements, C31 and any σv. Also, the group may be generated from any two σv planes. δ The group contains one set of symmetry planes σv intersecting in the principal (z) axis. The xz plane (but not the yz plane) is a member of that set. ε The lowest nonvanishing multipole moment in C31v is 2 (dipole 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 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.
| C29v | ||
| C30v | ||
| C31 | C31v | C31h D31 D31h D31d |
| C32v | ||
| C33v |
This Character Table for the C31v point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.