C62v E 2 C62 2 C31 2 C62^3 2 C31^2 2 C62^5 2 C31^3 2 C62^7 2 C31^4 2 C62^9 2 C31^5 2 C62^11 2 C31^6 2 C62^13 2 C31^7 2 C62^15 2 C31^8 2 C62^17 2 C31^9 2 C62^19 2 C31^10 2 C62^21 2 C31^11 2 C62^23 2 C31^12 2 C62^25 2 C31^13 2 C62^27 2 C31^14 2 C62^29 2 C31^15 C2 31 sv 31 sd <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 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 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 ... ..... ....... ......... ........... ............. B1 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 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. B2 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 1.00000 -1.00000 1.00000 -1.00000 -1.00000 1.00000 ... ... ..... ....... ......... ........... ............. E1 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 -0.10130 -0.30286 -0.50131 -0.69461 -0.88079 -1.05793 -1.22421 -1.37793 -1.51752 -1.64153 -1.74869 -1.83792 -1.90828 -1.95906 -1.98974 -2.00000 0.00000 0.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 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 -1.98974 -1.90828 -1.74869 -1.51752 -1.22421 -0.88079 -0.50131 -0.10130 0.30286 0.69461 1.05793 1.37793 1.64153 1.83792 1.95906 2.00000 0.00000 0.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 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 0.30286 0.88079 1.37793 1.74869 1.95906 1.98974 1.83792 1.51752 1.05793 0.50131 -0.10130 -0.69461 -1.22421 -1.64153 -1.90828 -2.00000 0.00000 0.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 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 1.95906 1.64153 1.05793 0.30286 -0.50131 -1.22421 -1.74869 -1.98974 -1.90828 -1.51752 -0.88079 -0.10130 0.69461 1.37793 1.83792 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 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 -0.50131 -1.37793 -1.90828 -1.95906 -1.51752 -0.69461 0.30286 1.22421 1.83792 1.98974 1.64153 0.88079 -0.10130 -1.05793 -1.74869 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 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 -1.90828 -1.22421 -0.10130 1.05793 1.83792 1.95906 1.37793 0.30286 -0.88079 -1.74869 -1.98974 -1.51752 -0.50131 0.69461 1.64153 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7 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 0.69461 1.74869 1.95906 1.22421 -0.10130 -1.37793 -1.98974 -1.64153 -0.50131 0.88079 1.83792 1.90828 1.05793 -0.30286 -1.51752 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 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 1.83792 0.69461 -0.88079 -1.90828 -1.74869 -0.50131 1.05793 1.95906 1.64153 0.30286 -1.22421 -1.98974 -1.51752 -0.10130 1.37793 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9 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 -0.88079 -1.95906 -1.51752 0.10130 1.64153 1.90828 0.69461 -1.05793 -1.98974 -1.37793 0.30286 1.74869 1.83792 0.50131 -1.22421 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 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 -1.74869 -0.10130 1.64153 1.83792 0.30286 -1.51752 -1.90828 -0.50131 1.37793 1.95906 0.69461 -1.22421 -1.98974 -0.88079 1.05793 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11 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 1.05793 1.98974 0.69461 -1.37793 -1.90828 -0.30286 1.64153 1.74869 -0.10130 -1.83792 -1.51752 0.50131 1.95906 1.22421 -0.88079 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 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 1.64153 -0.50131 -1.98974 -0.88079 1.37793 1.83792 -0.10130 -1.90828 -1.22421 1.05793 1.95906 0.30286 -1.74869 -1.51752 0.69461 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E13 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 -1.22421 -1.83792 0.30286 1.98974 0.69461 -1.64153 -1.51752 0.88079 1.95906 0.10130 -1.90828 -1.05793 1.37793 1.74869 -0.50131 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14 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 -1.51752 1.05793 1.83792 -0.50131 -1.98974 -0.10130 1.95906 0.69461 -1.74869 -1.22421 1.37793 1.64153 -0.88079 -1.90828 0.30286 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E15 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 1.37793 1.51752 -1.22421 -1.64153 1.05793 1.74869 -0.88079 -1.83792 0.69461 1.90828 -0.50131 -1.95906 0.30286 1.98974 -0.10130 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E16 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 1.37793 -1.51752 -1.22421 1.64153 1.05793 -1.74869 -0.88079 1.83792 0.69461 -1.90828 -0.50131 1.95906 0.30286 -1.98974 -0.10130 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E17 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 -1.51752 -1.05793 1.83792 0.50131 -1.98974 0.10130 1.95906 -0.69461 -1.74869 1.22421 1.37793 -1.64153 -0.88079 1.90828 0.30286 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E18 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 -1.22421 1.83792 0.30286 -1.98974 0.69461 1.64153 -1.51752 -0.88079 1.95906 -0.10130 -1.90828 1.05793 1.37793 -1.74869 -0.50131 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E19 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 1.64153 0.50131 -1.98974 0.88079 1.37793 -1.83792 -0.10130 1.90828 -1.22421 -1.05793 1.95906 -0.30286 -1.74869 1.51752 0.69461 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E20 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 1.05793 -1.98974 0.69461 1.37793 -1.90828 0.30286 1.64153 -1.74869 -0.10130 1.83792 -1.51752 -0.50131 1.95906 -1.22421 -0.88079 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E21 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 -1.74869 0.10130 1.64153 -1.83792 0.30286 1.51752 -1.90828 0.50131 1.37793 -1.95906 0.69461 1.22421 -1.98974 0.88079 1.05793 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E22 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 -0.88079 1.95906 -1.51752 -0.10130 1.64153 -1.90828 0.69461 1.05793 -1.98974 1.37793 0.30286 -1.74869 1.83792 -0.50131 -1.22421 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E23 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 1.83792 -0.69461 -0.88079 1.90828 -1.74869 0.50131 1.05793 -1.95906 1.64153 -0.30286 -1.22421 1.98974 -1.51752 0.10130 1.37793 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E24 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 0.69461 -1.74869 1.95906 -1.22421 -0.10130 1.37793 -1.98974 1.64153 -0.50131 -0.88079 1.83792 -1.90828 1.05793 0.30286 -1.51752 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E25 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 -1.90828 1.22421 -0.10130 -1.05793 1.83792 -1.95906 1.37793 -0.30286 -0.88079 1.74869 -1.98974 1.51752 -0.50131 -0.69461 1.64153 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E26 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 -0.50131 1.37793 -1.90828 1.95906 -1.51752 0.69461 0.30286 -1.22421 1.83792 -1.98974 1.64153 -0.88079 -0.10130 1.05793 -1.74869 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E27 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 1.95906 -1.64153 1.05793 -0.30286 -0.50131 1.22421 -1.74869 1.98974 -1.90828 1.51752 -0.88079 0.10130 0.69461 -1.37793 1.83792 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E28 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 0.30286 -0.88079 1.37793 -1.74869 1.95906 -1.98974 1.83792 -1.51752 1.05793 -0.50131 -0.10130 0.69461 -1.22421 1.64153 -1.90828 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E29 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 -1.98974 1.90828 -1.74869 1.51752 -1.22421 0.88079 -0.50131 0.10130 0.30286 -0.69461 1.05793 -1.37793 1.64153 -1.83792 1.95906 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E30 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 -0.10130 0.30286 -0.50131 0.69461 -0.88079 1.05793 -1.22421 1.37793 -1.51752 1.64153 -1.74869 1.83792 -1.90828 1.95906 -1.98974 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.989738646784 = 2*cos(2*π/62) = 2*cos(π/31) 1.959059882505 = 2*cos(4*π/62) = 2*cos(2*π/31) 1.908278512800 = 2*cos(6*π/62) = 2*cos(3*π/31) 1.837915623240 = 2*cos(8*π/62) = 2*cos(4*π/31) 1.748693232289 = 2*cos(10*π/62) = 2*cos(5*π/31) 1.641526882415 = 2*cos(12*π/62) = 2*cos(6*π/31) 1.517516245386 = 2*cos(14*π/62) = 2*cos(7*π/31) 1.377933838151 = 2*cos(16*π/62) = 2*cos(8*π/31) 1.224211965095 = 2*cos(18*π/62) = 2*cos(9*π/31) 1.057928020654 = 2*cos(20*π/62) = 2*cos(10*π/31) 0.880788303115 = 2*cos(22*π/62) = 2*cos(11*π/31) 0.694610505690 = 2*cos(24*π/62) = 2*cos(12*π/31) 0.501305064517 = 2*cos(26*π/62) = 2*cos(13*π/31) 0.302855555009 = 2*cos(28*π/62) = 2*cos(14*π/31) 0.101298337677 = 2*cos(30*π/62) = 2*cos(15*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 C62v point group is 124, and the order of the principal axis (C62) is 62. The group has 34 irreducible representations. β The C62v point group is isomorphic to D31d, D31h and D62. γ The C62v point group is generated by two symmetry elements, C62 and any σv (or, non-canonically, any σd). Also, the group may be generated from a σv plus a σd (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ There are two different sets of symmetry planes containing the principal axis (z axis in standard orientation). By convention, the set denoted as σv has the xz plane as a member, while the yz plane is a member of the σd set. ε The lowest nonvanishing multipole moment in C62v 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 much 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.
| C60v | ||
| C61v | ||
| C62 | C62v | C62h D62 D62h D62d S62 |
| C63v | ||
| C64v |
This Character Table for the C62v point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.