Character table for point group S8

=exp(2i/8)
S8 E S8 C4 (z) (S8)3 C2 (S8)5 (C4)3 (S8)7
linear functions,
rotations
quadratic
functions
cubic
functions
A +1 +1 +1 +1 +1 +1 +1 +1 Rz x2+y2, z2 -
B +1 -1 +1 -1 +1 -1 +1 -1 z - z3, z(x2+y2)
E1 +1
+1
+
+*
+i
-i
-*
-
-1
-1
-
-*
-i
+i
+*
+
x+iy
x-iy
- (xz2, yz2) [x(x2+y2), y(x2+y2)]
E2 +1
+1
+i
-i
-1
-1
-i
+i
+1
+1
+i
-i
-1
-1
-i
+i
- (x2-y2, xy) [xyz, z(x2-y2)]
E3 +1
+1
-
-*
+i
-i
+*
+
-1
-1
+
+*
-i
+i
-*
-
Rx+iRy
Rx-iRy
(xz, yz) [y(3x2-y2), x(x2-3y2)]


Additional information

Number of symmetry elements h = 8
Number of irreducible representations n = 8
Number of real irreducible representations n = 5
Abelian group yes
Subgroups C2 , C4
Optical Isomerism (Chirality) no


Reduction formula for point group S8

Type of representation

Information for point groups with complex irreducible representations

general 3N vib

E S8 C4 (z) (S8)3 C2 (S8)5 (C4)3 (S8)7




Multipoles

dipole (p) B+E1
quadrupole (d) A+E2+E3
octopole (f) B+E1+E2+E3
hexadecapole (g) A+2B+E1+E2+E3
32-pole (h) 2A+B+E1+E2+2E3
64-pole (i) A+2B+2E1+2E2+E3
128-pole (j) 2A+B+2E1+2E2+2E3
256-pole(k) 3A+2B+2E1+2E2+2E3
512-pole (l) 2A+3B+3E1+2E2+2E3

First nonvanishing multipole: quadrupole

Literature



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