C48v E 2 C48 2 C24 2 C16 2 C12 2 C48^5 2 C8 2 C48^7 2 C6 2 C16^3 2 C24^5 2 C48^11 2 C4 2 C48^13 2 C24^7 2 C16^5 2 C3 2 C48^17 2 C8^3 2 C48^19 2 C12^5 2 C16^7 2 C24^11 2 C48^23 C2 24 sv 24 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 ... ..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 ..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 ... ... ..... ....... ......... ........... ............. 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 ... ... ..... ....... ......... ........... ............. E1 2.00000 1.98289 1.93185 1.84776 1.73205 1.58671 1.41421 1.21752 1.00000 0.76537 0.51764 0.26105 0.00000 -0.26105 -0.51764 -0.76537 -1.00000 -1.21752 -1.41421 -1.58671 -1.73205 -1.84776 -1.93185 -1.98289 -2.00000 0.00000 0.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.00000 1.93185 1.73205 1.41421 1.00000 0.51764 0.00000 -0.51764 -1.00000 -1.41421 -1.73205 -1.93185 -2.00000 -1.93185 -1.73205 -1.41421 -1.00000 -0.51764 0.00000 0.51764 1.00000 1.41421 1.73205 1.93185 2.00000 0.00000 0.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -1.84776 -1.41421 -0.76537 0.00000 0.76537 1.41421 1.84776 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 0.00000 0.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 2.00000 1.73205 1.00000 0.00000 -1.00000 -1.73205 -2.00000 -1.73205 -1.00000 0.00000 1.00000 1.73205 2.00000 1.73205 1.00000 0.00000 -1.00000 -1.73205 -2.00000 -1.73205 -1.00000 0.00000 1.00000 1.73205 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 2.00000 1.58671 0.51764 -0.76537 -1.73205 -1.98289 -1.41421 -0.26105 1.00000 1.84776 1.93185 1.21752 0.00000 -1.21752 -1.93185 -1.84776 -1.00000 0.26105 1.41421 1.98289 1.73205 0.76537 -0.51764 -1.58671 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.21752 -0.51764 -1.84776 -1.73205 -0.26105 1.41421 1.98289 1.00000 -0.76537 -1.93185 -1.58671 0.00000 1.58671 1.93185 0.76537 -1.00000 -1.98289 -1.41421 0.26105 1.73205 1.84776 0.51764 -1.21752 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 1.00000 -1.00000 -2.00000 -1.00000 1.00000 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9 2.00000 0.76537 -1.41421 -1.84776 0.00000 1.84776 1.41421 -0.76537 -2.00000 -0.76537 1.41421 1.84776 0.00000 -1.84776 -1.41421 0.76537 2.00000 0.76537 -1.41421 -1.84776 0.00000 1.84776 1.41421 -0.76537 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 0.51764 -1.73205 -1.41421 1.00000 1.93185 0.00000 -1.93185 -1.00000 1.41421 1.73205 -0.51764 -2.00000 -0.51764 1.73205 1.41421 -1.00000 -1.93185 0.00000 1.93185 1.00000 -1.41421 -1.73205 0.51764 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 0.26105 -1.93185 -0.76537 1.73205 1.21752 -1.41421 -1.58671 1.00000 1.84776 -0.51764 -1.98289 0.00000 1.98289 0.51764 -1.84776 -1.00000 1.58671 1.41421 -1.21752 -1.73205 0.76537 1.93185 -0.26105 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 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 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E13 2.00000 -0.26105 -1.93185 0.76537 1.73205 -1.21752 -1.41421 1.58671 1.00000 -1.84776 -0.51764 1.98289 0.00000 -1.98289 0.51764 1.84776 -1.00000 -1.58671 1.41421 1.21752 -1.73205 -0.76537 1.93185 0.26105 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 -0.51764 -1.73205 1.41421 1.00000 -1.93185 0.00000 1.93185 -1.00000 -1.41421 1.73205 0.51764 -2.00000 0.51764 1.73205 -1.41421 -1.00000 1.93185 0.00000 -1.93185 1.00000 1.41421 -1.73205 -0.51764 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E15 2.00000 -0.76537 -1.41421 1.84776 0.00000 -1.84776 1.41421 0.76537 -2.00000 0.76537 1.41421 -1.84776 0.00000 1.84776 -1.41421 -0.76537 2.00000 -0.76537 -1.41421 1.84776 0.00000 -1.84776 1.41421 0.76537 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E16 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E17 2.00000 -1.21752 -0.51764 1.84776 -1.73205 0.26105 1.41421 -1.98289 1.00000 0.76537 -1.93185 1.58671 0.00000 -1.58671 1.93185 -0.76537 -1.00000 1.98289 -1.41421 -0.26105 1.73205 -1.84776 0.51764 1.21752 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E18 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E19 2.00000 -1.58671 0.51764 0.76537 -1.73205 1.98289 -1.41421 0.26105 1.00000 -1.84776 1.93185 -1.21752 0.00000 1.21752 -1.93185 1.84776 -1.00000 -0.26105 1.41421 -1.98289 1.73205 -0.76537 -0.51764 1.58671 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E20 2.00000 -1.73205 1.00000 0.00000 -1.00000 1.73205 -2.00000 1.73205 -1.00000 0.00000 1.00000 -1.73205 2.00000 -1.73205 1.00000 0.00000 -1.00000 1.73205 -2.00000 1.73205 -1.00000 0.00000 1.00000 -1.73205 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E21 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 1.84776 -1.41421 0.76537 0.00000 -0.76537 1.41421 -1.84776 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E22 2.00000 -1.93185 1.73205 -1.41421 1.00000 -0.51764 0.00000 0.51764 -1.00000 1.41421 -1.73205 1.93185 -2.00000 1.93185 -1.73205 1.41421 -1.00000 0.51764 0.00000 -0.51764 1.00000 -1.41421 1.73205 -1.93185 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E23 2.00000 -1.98289 1.93185 -1.84776 1.73205 -1.58671 1.41421 -1.21752 1.00000 -0.76537 0.51764 -0.26105 0.00000 0.26105 -0.51764 0.76537 -1.00000 1.21752 -1.41421 1.58671 -1.73205 1.84776 -1.93185 1.98289 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.982889722748 = 2*cos(2*π/48) = 2*cos(π/24) = √2+√2+√3 1.931851652578 = 2*cos(4*π/48) = 2*cos(π/12) = √2+√3 1.847759065023 = 2*cos(6*π/48) = 2*cos(π/8) = √2+√2 1.732050807569 = 2*cos(8*π/48) = 2*cos(π/6) = √3 1.586706680582 = 2*cos(10*π/48) = 2*cos(5*π/24) = √2+√2−√3 1.414213562373 = 2*cos(12*π/48) = 2*cos(π/4) = √2 1.217522858017 = 2*cos(14*π/48) = 2*cos(7*π/24) = √2−√2−√3 0.765366864730 = 2*cos(18*π/48) = 2*cos(3*π/8) = √2−√2 0.517638090205 = 2*cos(20*π/48) = 2*cos(5*π/12) = √2−√3 0.261052384440 = 2*cos(22*π/48) = 2*cos(11*π/24) = √2−√2+√3 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 C48v point group is 96, and the order of the principal axis (C48) is 48. The group has 27 irreducible representations. β The C48v point group is isomorphic to D24d and D48. γ The C48v point group is generated by two symmetry elements, C48 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 C48v 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. θ The point group corresponds to a constructible polygon, as the order of the principal axis is a product of any number of different Fermat primes (3,5,17,257,65537) times an arbitrary power of two. Therefore, all characters have an algebraic degree which is a power of two and can be expressed as radicals involving only square roots and integer numbers.
| C46v | ||
| C47v | ||
| C48 | C48v | C48h D48 D48h D48d S48 |
| C49v | ||
| C50v |
This Character Table for the C48v point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.