Character Tables for Common Symmetry Groups - The Dnh

D2hEC2(z)C2(y)C2(x)iσ(xy)σ(xz)σ(yz)  
Ag 1 1 1 1 1 1 1 1   x2, y2, z2
B1g 1 1 -1 -1 1 1 -1 -1 Rz xy
B2g 1 -1 1 -1 1 -1 1 -1 Ry xz
B3g 1 -1 -1 1 1 -1 -1 1 Rx yz
Au 1 1 1 1 -1 -1 -1 -1    
B1u 1 1 -1 -1 -1 -1 1 1 z  
B2u 1 -1 1 -1 -1 1 -1 1 y  
B3u 1 -1 -1 1 -1 1 1 -1 x  

 

D3hE2C33C2σh2S33σv  
A1' 1 1 1 1 1 1   x2 + y2, z2
A2' 1 1 -1 1 1 -1 Rz  
E 2 -1 0 2 -1 0 (x, y) (x2 - y2, xy)
A1" 1 1 1 -1 -1 -1    
A2" 1 1 -1 -1 -1 1 z  
E' 2 -1 0 -2 1 0 (Rx, Ry) (xz, yz)


2σd

D4hE2C4C22C2'2C2"i2S4σh2σv  
A1g 1 1 1 1 1 1 1 1 1 1   x2 + y2, z2
A2g 1 1 1 -1 -1 1 1 1 -1 -1 Rz  
B1g 1 -1 1 1 -1 1 -1 1 1 -1   x2 - y2
B2g 1 -1 1 -1 1 1 -1 1 -1 1   xy
Eg 2 0 -2 0 0 2 0 -2 0 0 (Rx, Ry) (xz, yz)
A1u 1 1 1 1 1 -1 -1 -1 -1 -1    
A2u 1 1 1 -1 -1 -1 -1 -1 1 1 z  
B1u 1 -1 1 1 -1 -1 1 -1 -1 1    
B2u 1 -1 1 -1 1 -1 1 -1 1 1    
Eu 2 0 -2 0 0 -2 0 2 0 0 (x, y)  

 

D5hE2C52C525C2σh2S52S535σv  
A1' 1 1 1 1 1 1 1 1   x2 + y2, z2
A2' 1 1 1 -1 1 1 1 -1 Rz  
E1 2 2cos72° 2cos144° 0 2 2cos72° 2cos144° 0 (x, y)  
E2 2 2cos144° 2cos72° 0 2 2cos144° 2cos72° 0   (x2 - y2, xy)
A1" 1 1 1 1 -1 -1 -1 -1    
A2" 1 1 1 -1 -1 -1 -1 1 z  
E1" 2 2cos72° 2cos144° 0 -2 -2cos72° -2cos144° 0 (Rx, Ry) (xz, yz)
E2" 2 2cos144° 2cos72° 0 -2 -2cos144° -2cos72° 0    

 

D6hE2C62C3C23C2'3C2'i2S32S6σh3σd3σv  
A1g 1 1 1 1 1 1 1 1 1 1 1 1   x2 + y2, z2
A2g 1 1 1 1 -1 -1 1 1 1 1 -1 -1 Rz  
B1g 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1    
B2g 1 -1 1 -1 -1 1 1 -1 1 -1 -1 1    
E1g 2 1 -1 -2 0 0 2 1 -1 -2 0 0 (Rx, Ry) (xz, yz)
E2g 2 -1 -1 2 0 0 2 -1 -1 2 0 0   (x2 - y2, xy)
A1u 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1    
A2u 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 z  
B1u 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1    
B2u 1 -1 1 -1 -1 1 -1 1 -1 1 1 -1    
E1u 2 1 -1 -2 0 0 -2 -1 1 2 0 0 (x, y)  
E2u 2 -1 -1 2 0 0 -2 1 1 -2 0 0    

Reference: F.A. Cotton, "Chemical Applications of Group Theory," 3rd ed., Wiley-Interscience, NY, 1990.