Character table for the C31h point group

C31h     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 sh       2 S31    2 S31^3  2 S31^5  2 S31^7  2 S31^9  2 S31^11 2 S31^13 2 S31^15 2 S31^17 2 S31^19 2 S31^21 2 S31^23 2 S31^25 2 S31^27 2 S31^29   <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A'      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
A"      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 .............
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  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     ... TT. ..... ....TT. ......... ........TT. .............
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 -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     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  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     ... ... 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 -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     ... ... ..... ..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  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     ... ... ..... 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 -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     ... ... ..... ....... ..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  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     ... ... ..... ....... 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 -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     ... ... ..... ....... ......... ..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  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     ... ... ..... ....... ......... 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 -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     ... ... ..... ....... ......... ........... ..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  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     ... ... ..... ....... ......... ........... 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 -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     ... ... ..... ....... ......... ........... .............
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  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     ... ... ..... ....... ......... ........... .............
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 -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     ... ... ..... ....... ......... ........... .............
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  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     ... ... ..... ....... ......... ........... .............
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 -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     ... ... ..... ....... ......... ........... .............
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  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     ... ... ..... ....... ......... ........... .............
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 -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     ... ... ..... ....... ......... ........... .............
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  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     ... ... ..... ....... ......... ........... .............
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 -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     ... ... ..... ....... ......... ........... .............
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  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     ... ... ..... ....... ......... ........... .............
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 -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     ... ... ..... ....... ......... ........... .............
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  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     ... ... ..... ....... ......... ........... .............
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 -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     ... ... ..... ....... ......... ........... .............
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  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     ... ... ..... ....... ......... ........... .............
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 -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     ... ... ..... ....... ......... ........... .............
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  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     ... ... ..... ....... ......... ........... .............
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 -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     ... ... ..... ....... ......... ........... .............
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  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     ... ... ..... ....... ......... ........... .............
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 -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     ... ... ..... ....... ......... ........... .............

 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

A'   R+d+g+i+k+m  Rz, z2, z4, z6 
A"   p+f+h+j+l    z, z3, z5 
E1'  p+f+h+j+l    {x, y}, {xz2, yz2}, {xz4, yz4} 
E1"  R+d+g+i+k+m  {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} 
E2'  d+g+i+k+m    {x2y2, xy}, {z2(x2y2), xyz2}, {z4(x2y2), xyz4} 
E2"  f+h+j+l      {z(x2y2), xyz}, {z3(x2y2), xyz3} 
E3'  f+h+j+l      {x(x2−3y2), y(3x2y2)}, {xz2(x2−3y2), yz2(3x2y2)} 
E3"  g+i+k+m      {xz(x2−3y2), yz(3x2y2)}, {xz3(x2−3y2), yz3(3x2y2)} 
E4'  g+i+k+m      {(x2y2)2−4x2y2, xy(x2y2)}, {z2((x2y2)2−4x2y2), xyz2(x2y2)} 
E4"  h+j+l        {z((x2y2)2−4x2y2), xyz(x2y2)} 
E5'  h+j+l        {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2y2)((5−2√5)x2y2)} 
E5"  i+k+m        {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2y2)((5−2√5)x2y2)} 
E6'  i+k+m        {x2(x2−3y2)2y2(3x2y2)2, xy(x2−3y2)(3x2y2)} 
E6"  j+l 
E7'  j+l 
E7"  k+m 
E8'  k+m 
E8"  l 
E9'  l 
E9"  m 
E10' m 

 Notes:

    α  The order of the C31h point group is 62, and the order of the principal axis (S31) is 62. The group has 32 irreducible representations.

    β  The C31h point group could also be named S31, as it contains the S31 axis as its only symmetry element.
       Another rare designation is C62i because the S31 axis is identical to a roto-inversion axis of order 62.

    γ  The C31h point group is isomorphic to C62 and S62.

    δ  The C31h point group is generated by one single symmetry element, S31. Therefore, it is a cyclic group.
       The canonical choice, however, is to use redundant generators: C31 and σh.

    ε  The lowest nonvanishing multipole moment in C31h is 4 (quadrupole moment).

    ζ  This is an Abelian point group (the commutative law holds between all symmetry operations).
       The C31h group is Abelian because it contains only one symmetry element, all the powers of which necessarily commute (sufficient condition).
       In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional.

    η  Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters.
       These 60 cases have been combined into 30 two-dimensional representations that are no longer irreducible but have real-valued characters.
       Accordingly, 30 pairs of left and right rotations have been combined into one two-membered pseudo-class each.

    θ  The 30 reducible “E” representations almost behave like true irreducible representations.
       Their norm, however, is twice the group order. Therefore, they have been marked with an asterisk in the table.
       This is essential when trying to decompose a reducible representation into “irreducible” ones using the familiar projection formula.

    ι  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 C31h point group was created by Gernot Katzer.

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