C44v E 2 C44 2 C22 2 C44^3 2 C11 2 C44^5 2 C22^3 2 C44^7 2 C11^2 2 C44^9 2 C22^5 2 C4 2 C11^3 2 C44^13 2 C22^7 2 C44^15 2 C11^4 2 C44^17 2 C22^9 2 C44^19 2 C11^5 2 C44^21 C2 22 sv 22 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 ... ..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 ..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 ... ... ..... ....... ......... ........... ............. 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 ... ... ..... ....... ......... ........... ............. E1 2.00000 1.97964 1.91899 1.81926 1.68251 1.51150 1.30972 1.08128 0.83083 0.56347 0.28463 0.00000 -0.28463 -0.56347 -0.83083 -1.08128 -1.30972 -1.51150 -1.68251 -1.81926 -1.91899 -1.97964 -2.00000 0.00000 0.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.00000 1.91899 1.68251 1.30972 0.83083 0.28463 -0.28463 -0.83083 -1.30972 -1.68251 -1.91899 -2.00000 -1.91899 -1.68251 -1.30972 -0.83083 -0.28463 0.28463 0.83083 1.30972 1.68251 1.91899 2.00000 0.00000 0.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.00000 1.81926 1.30972 0.56347 -0.28463 -1.08128 -1.68251 -1.97964 -1.91899 -1.51150 -0.83083 0.00000 0.83083 1.51150 1.91899 1.97964 1.68251 1.08128 0.28463 -0.56347 -1.30972 -1.81926 -2.00000 0.00000 0.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463 0.83083 1.68251 2.00000 1.68251 0.83083 -0.28463 -1.30972 -1.91899 -1.91899 -1.30972 -0.28463 0.83083 1.68251 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 2.00000 1.51150 0.28463 -1.08128 -1.91899 -1.81926 -0.83083 0.56347 1.68251 1.97964 1.30972 0.00000 -1.30972 -1.97964 -1.68251 -0.56347 0.83083 1.81926 1.91899 1.08128 -0.28463 -1.51150 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 2.00000 1.30972 -0.28463 -1.68251 -1.91899 -0.83083 0.83083 1.91899 1.68251 0.28463 -1.30972 -2.00000 -1.30972 0.28463 1.68251 1.91899 0.83083 -0.83083 -1.91899 -1.68251 -0.28463 1.30972 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.08128 -0.83083 -1.97964 -1.30972 0.56347 1.91899 1.51150 -0.28463 -1.81926 -1.68251 0.00000 1.68251 1.81926 0.28463 -1.51150 -1.91899 -0.56347 1.30972 1.97964 0.83083 -1.08128 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 1.68251 -0.28463 -1.91899 -1.30972 0.83083 2.00000 0.83083 -1.30972 -1.91899 -0.28463 1.68251 1.68251 -0.28463 -1.91899 -1.30972 0.83083 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9 2.00000 0.56347 -1.68251 -1.51150 0.83083 1.97964 0.28463 -1.81926 -1.30972 1.08128 1.91899 0.00000 -1.91899 -1.08128 1.30972 1.81926 -0.28463 -1.97964 -0.83083 1.51150 1.68251 -0.56347 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 0.28463 -1.91899 -0.83083 1.68251 1.30972 -1.30972 -1.68251 0.83083 1.91899 -0.28463 -2.00000 -0.28463 1.91899 0.83083 -1.68251 -1.30972 1.30972 1.68251 -0.83083 -1.91899 0.28463 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 -1.30972 1.68251 0.83083 -1.91899 -0.28463 2.00000 -0.28463 -1.91899 0.83083 1.68251 -1.30972 -1.30972 1.68251 0.83083 -1.91899 -0.28463 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E13 2.00000 -0.56347 -1.68251 1.51150 0.83083 -1.97964 0.28463 1.81926 -1.30972 -1.08128 1.91899 0.00000 -1.91899 1.08128 1.30972 -1.81926 -0.28463 1.97964 -0.83083 -1.51150 1.68251 0.56347 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 -0.83083 -1.30972 1.91899 -0.28463 -1.68251 1.68251 0.28463 -1.91899 1.30972 0.83083 -2.00000 0.83083 1.30972 -1.91899 0.28463 1.68251 -1.68251 -0.28463 1.91899 -1.30972 -0.83083 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E15 2.00000 -1.08128 -0.83083 1.97964 -1.30972 -0.56347 1.91899 -1.51150 -0.28463 1.81926 -1.68251 0.00000 1.68251 -1.81926 0.28463 1.51150 -1.91899 0.56347 1.30972 -1.97964 0.83083 1.08128 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E16 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 -1.30972 -0.28463 1.68251 -1.91899 0.83083 0.83083 -1.91899 1.68251 -0.28463 -1.30972 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E17 2.00000 -1.51150 0.28463 1.08128 -1.91899 1.81926 -0.83083 -0.56347 1.68251 -1.97964 1.30972 0.00000 -1.30972 1.97964 -1.68251 0.56347 0.83083 -1.81926 1.91899 -1.08128 -0.28463 1.51150 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E18 2.00000 -1.68251 0.83083 0.28463 -1.30972 1.91899 -1.91899 1.30972 -0.28463 -0.83083 1.68251 -2.00000 1.68251 -0.83083 -0.28463 1.30972 -1.91899 1.91899 -1.30972 0.28463 0.83083 -1.68251 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E19 2.00000 -1.81926 1.30972 -0.56347 -0.28463 1.08128 -1.68251 1.97964 -1.91899 1.51150 -0.83083 0.00000 0.83083 -1.51150 1.91899 -1.97964 1.68251 -1.08128 0.28463 0.56347 -1.30972 1.81926 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E20 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 -1.91899 1.68251 -1.30972 0.83083 -0.28463 -0.28463 0.83083 -1.30972 1.68251 -1.91899 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E21 2.00000 -1.97964 1.91899 -1.81926 1.68251 -1.51150 1.30972 -1.08128 0.83083 -0.56347 0.28463 0.00000 -0.28463 0.56347 -0.83083 1.08128 -1.30972 1.51150 -1.68251 1.81926 -1.91899 1.97964 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.979642883762 = 2*cos(2*π/44) = 2*cos(π/22) 1.918985947229 = 2*cos(4*π/44) = 2*cos(π/11) 1.819263990709 = 2*cos(6*π/44) = 2*cos(3*π/22) 1.682507065662 = 2*cos(8*π/44) = 2*cos(2*π/11) 1.511499148709 = 2*cos(10*π/44) = 2*cos(5*π/22) 1.309721467891 = 2*cos(12*π/44) = 2*cos(3*π/11) 1.081281634911 = 2*cos(14*π/44) = 2*cos(7*π/22) 0.830830026004 = 2*cos(16*π/44) = 2*cos(4*π/11) 0.563465113683 = 2*cos(18*π/44) = 2*cos(9*π/22) 0.284629676547 = 2*cos(20*π/44) = 2*cos(5*π/11) 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 C44v point group is 88, and the order of the principal axis (C44) is 44. The group has 25 irreducible representations. β The C44v point group is isomorphic to D22d and D44. γ The C44v point group is generated by two symmetry elements, C44 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 contains both the xz and the yz planes. ε The lowest nonvanishing multipole moment in C44v 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.
| C42v | ||
| C43v | ||
| C44 | C44v | C44h D44 D44h D44d S44 |
| C45v | ||
| C46v |
This Character Table for the C44v point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.