Character table for the D31d point group

D31d     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 C2'   i        2 S62    2 S62^3  2 S62^5  2 S62^7  2 S62^9  2 S62^11 2 S62^13 2 S62^15 2 S62^17 2 S62^19 2 S62^21 2 S62^23 2 S62^25 2 S62^27 2 S62^29 31 sd      <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A1g     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
A2g     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 ... ..... ....... ......... ........... .............
E1g     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  2.00000 -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  0.00000     TT. ... ..TT. ....... ......TT. ........... ..........TT.
E2g     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  2.00000  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  0.00000     ... ... TT... ....... ....TT... ........... ........TT...
E3g     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  2.00000 -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  0.00000     ... ... ..... ....... ..TT..... ........... ......TT.....
E4g     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  2.00000  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  0.00000     ... ... ..... ....... TT....... ........... ....TT.......
E5g     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  2.00000 -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  0.00000     ... ... ..... ....... ......... ........... ..TT.........
E6g     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  2.00000  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  0.00000     ... ... ..... ....... ......... ........... TT...........
E7g     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  2.00000 -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  0.00000     ... ... ..... ....... ......... ........... .............
E8g     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  2.00000  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  0.00000     ... ... ..... ....... ......... ........... .............
E9g     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  2.00000 -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  0.00000     ... ... ..... ....... ......... ........... .............
E10g    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  2.00000  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  0.00000     ... ... ..... ....... ......... ........... .............
E11g    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  2.00000 -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  0.00000     ... ... ..... ....... ......... ........... .............
E12g    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  2.00000  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  0.00000     ... ... ..... ....... ......... ........... .............
E13g    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  2.00000 -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  0.00000     ... ... ..... ....... ......... ........... .............
E14g    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  2.00000  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  0.00000     ... ... ..... ....... ......... ........... .............
E15g    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  2.00000 -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  0.00000     ... ... ..... ....... ......... ........... .............
A1u     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     ... ... ..... ....... ......... ........... .............
A2u     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 .............
E1u     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 -2.00000  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  0.00000     ... TT. ..... ....TT. ......... ........TT. .............
E2u     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 -2.00000 -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  0.00000     ... ... ..... ..TT... ......... ......TT... .............
E3u     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 -2.00000  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  0.00000     ... ... ..... TT..... ......... ....TT..... .............
E4u     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 -2.00000 -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  0.00000     ... ... ..... ....... ......... ..TT....... .............
E5u     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 -2.00000  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  0.00000     ... ... ..... ....... ......... TT......... .............
E6u     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 -2.00000 -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  0.00000     ... ... ..... ....... ......... ........... .............
E7u     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 -2.00000  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  0.00000     ... ... ..... ....... ......... ........... .............
E8u     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 -2.00000 -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  0.00000     ... ... ..... ....... ......... ........... .............
E9u     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 -2.00000  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  0.00000     ... ... ..... ....... ......... ........... .............
E10u    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 -2.00000 -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  0.00000     ... ... ..... ....... ......... ........... .............
E11u    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 -2.00000  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  0.00000     ... ... ..... ....... ......... ........... .............
E12u    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 -2.00000 -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  0.00000     ... ... ..... ....... ......... ........... .............
E13u    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 -2.00000  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  0.00000     ... ... ..... ....... ......... ........... .............
E14u    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 -2.00000 -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  0.00000     ... ... ..... ....... ......... ........... .............
E15u    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 -2.00000  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  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

A1g  d+g+i+k+m    z2, z4, z6 
A2g  R            Rz 
E1g  R+d+g+i+k+m  {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} 
E2g  d+g+i+k+m    {x2y2, xy}, {z2(x2y2), xyz2}, {z4(x2y2), xyz4} 
E3g  g+i+k+m      {xz(x2−3y2), yz(3x2y2)}, {xz3(x2−3y2), yz3(3x2y2)} 
E4g  g+i+k+m      {(x2y2)2−4x2y2, xy(x2y2)}, {z2((x2y2)2−4x2y2), xyz2(x2y2)} 
E5g  i+k+m        {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2y2)((5−2√5)x2y2)} 
E6g  i+k+m        {x2(x2−3y2)2y2(3x2y2)2, xy(x2−3y2)(3x2y2)} 
E7g  k+m 
E8g  k+m 
E9g  m 
E10g m 
A2u  p+f+h+j+l    z, z3, z5 
E1u  p+f+h+j+l    {x, y}, {xz2, yz2}, {xz4, yz4} 
E2u  f+h+j+l      {z(x2y2), xyz}, {z3(x2y2), xyz3} 
E3u  f+h+j+l      {x(x2−3y2), y(3x2y2)}, {xz2(x2−3y2), yz2(3x2y2)} 
E4u  h+j+l        {z((x2y2)2−4x2y2), xyz(x2y2)} 
E5u  h+j+l        {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2y2)((5−2√5)x2y2)} 
E6u  j+l 
E7u  j+l 
E8u  l 
E9u  l 

 Notes:

    α  The order of the D31d point group is 124, and the order of the principal axis (S62) is 62. The group has 34 irreducible representations.

    β  The D31d point group is isomorphic to D31h, C62v and D62.

    γ  The D31d point group is generated by two symmetry elements, S62 and either a perpendicular C2 or a vertical σd.
       Also, the group may be generated from any C2 plus any σd plane.
       The canonical choice, however, is to use redundant generators: C31, C2 and i.

    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.
       Reversely, the single set of symmetry planes denoted σd contains the yz plane but not the xz plane.

    ε  The lowest nonvanishing multipole moment in D31d 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  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.

This Character Table for the D31d point group was created by Gernot Katzer.

For other groups and some explanations, see the Main Page.