Character table for point group Th

=exp(2i/3)
Th E 4C3 4(C3)2 3C2 i 4(S6)5 4S6 3h
linear functions,
rotations
quadratic
functions
cubic
functions
Ag +1 +1 +1 +1 +1 +1 +1 +1 - x2+y2+z2 -
Eg +1
+1
+
+*
+*
+
+1
+1
+1
+1
+
+*
+*
+
+1
+1
- (x2-y2, 2z2-x2-y2) -
Tg +3 0 0 -1 +3 0 0 -1 (Rx, Ry, Rz) (xy, xz, yz) -
Au +1 +1 +1 +1 -1 -1 -1 -1 - - xyz
Eu +1
+1
+
+*
+*
+
+1
+1
-1
-1
-
-*
-*
-
-1
-1
- - -
Tu +3 0 0 -1 -3 0 0 +1 (x, y, z) - (x3, y3, z3) (xy2, x2z, yz2) (xz2, x2y, y2z)


Additional information

Number of symmetry elements h = 24
Number of irreducible representations n = 8
Number of real irreducible representations n = 6
Abelian group no
Number of subgroups10
Subgroups Cs , Ci , C2 , C3 , D2 , C2v , C2h , D2h , S6 , T
Optical Isomerism (Chirality) no


Reduction formula for point group Th

Type of representation

Information for point groups with complex irreducible representations

general 3N vib

E 4C3 4(C3)2 3C2 i 4(S6)5 4S6 3h




Examples

[Mg(H2O)6]2+-Ion Bromofullerene C60Br24



Multipoles

dipole (p) Tu
quadrupole (d) Eg+Tg
octopole (f) Au+2Tu
hexadecapole (g) Ag+Eg+2Tg
32-pole (h) Eu+3Tu
64-pole (i) 2Ag+Eg+3Tg
128-pole (j) Au+Eu+4Tu
256-pole(k) Ag+2Eg+4Tg
512-pole (l) 2Au+Eu+5Tu

First nonvanishing multipole: hexadecapole

Literature



Character tables for chemically important point groups Computational Laboratory for Analysis, Modeling and Visualization Jacobs University Bremen