Character table for point group S12

=exp(2i/12)
S12 E S12 C6 S4 C3 (S12)5 C2 (S12)7 (C3)2 (S4)3 (C6)5 (S12)11
linear functions,
rotations
quadratic
functions
cubic
functions
A +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 Rz z2, x2+y2 -
B +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 z - z3, z(x2+y2)
E1 +1
+1
+
+*
+2
+2*
+i
-i
-2*
-2
-*
-
-1
-1
-
-*
-2
-2*
-i
+i
+2*
+2
+*
+
x+iy
x-iy
- (xz2, yz2) [x(x2+y2), y(x2+y2)]
E2 +1
+1
+2
+2*
-2*
+2
-1
-1
-2
-2*
+2*
+2
+1
+1
+2
+2*
-2*
-2
-1
-1
-2
-2*
+2*
+2
- (x2-y2, xy) -
E3 +1
+1
+i
-i
-1
-1
-i
+i
+1
+1
+i
-i
-1
-1
-i
+i
+1
+1
+i
-i
-1
-1
-i
+i
- - [x(x2-3y2), y(3x2-y2)]
E4 +1
+1
-2*
-2
-2
-2*
+1
+1
-2*
-2
-2
-2*
+1
+1
-2*
-2
-2
-2*
+1
+1
-2*
-2
-2
-2*
- - [xyz, z(x2-y2)]
E5 +1
+1
-*
+
+2*
+2
+i
-i
-2
+2*
+
+*
-1
-1
+*
+
-2*
-2
-i
+i
+2
+2*
-
-*
Rx-iRy
Rx+iRy
(xz, yz) -


Additional information

Number of symmetry elements h = 12
Number of irreducible representations n = 12
Number of real irreducible representations n = 7
Abelian group yes
Number of subgroups4
Subgroups C2 , C3 , C6 , S4
Optical Isomerism (Chirality) no


Reduction formula for point group S12

Type of representation

Information for point groups with complex irreducible representations

general 3N vib

E S12 C6 S4 C3 (S12)5 C2 (S12)7 (C3)2 (S4)3 (C6)5 (S12)11




Multipoles

dipole (p) B+E1
quadrupole (d) A+E2+E5
octopole (f) B+E1+E3+E4
hexadecapole (g) A+E2+E3+E4+E5
32-pole (h) B+E1+E2+E3+E4+E5
64-pole (i) A+2B+E1+E2+E3+E4+E5
128-pole (j) 2A+B+E1+E2+E3+E4+2E5
256-pole(k) A+2B+2E1+E2+E3+2E4+E5
512-pole (l) 2A+B+E1+2E2+2E3+E4+2E5

First nonvanishing multipole: quadrupole

Literature



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