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Character Table

$C_1$ $E$
$A$ 1


$C_s$ $E$ $\sigma_h$    
$A$ 1 1 $x, y, R_z$ $x^2, y^2, z^2, xy$
$B$ 1 $-1$ $z, R_x, R_y$ $yz, xz$


$C_i$ $E$ $i$    
$A_g$ 1 1 $R_x, R_y, R_z$ $x^2, y^2, z^2, xy, xz, yz$
$A_u$ 1 $-1$ $x, y, z$  


$C_2$ $E$ $C_2$    
$A$ 1 1 $z, R_z$ $x^2, y^2, z^2, xy$
$B$ 1 $-1$ $x, y, R_x, R_y$ $yz, xz$


$C_3$ $E$ $C_3$ ${C_3}^2$   $\varepsilon=\mathop{\rm exp}\nolimits (2\pi i/3)$
$A$ 1 1 1 $z, R_z$ $x^2, y^2, z^2, xy$
$E$ $\cases{1\cr 1\cr}$ $\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $\left.\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $(x,y)(R_x,R_y)$ $(x^2-y^2,xy)(yz,xz)$


$C_4$ $E$ $C_3$ $C_2$ ${C_4}^3$    
$A$ 1 1 1 1 $z, R_z$ $x^2+y^2, z^2$
$B$ 1 $-1$ 1 $-1$   $x^2-y^2, xy$
$E$ $\cases{\hfill 1\cr \hfill 1\cr}$ $\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $\left.\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $(x,y)(R_x,R_y)$ $(yz,xz)$


$C_5$ $E$ $C_5$ ${C_5}^2$ ${C_5}^3$ ${C_5}^4$   $\varepsilon=\mathop{\rm exp}\nolimits (2\pi i/5)$
$A$ 1 1 1 1 1 $z, R_z$ $x^2+y^2, z^2$
$E_1$ $\cases{\hfill 1\cr \hfill 1\cr}$ $\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $\left.\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $(x,y)(R_x,R_y)$ $(yz,xz)$
$E_2$ $\cases{1\cr 1\cr}$ $\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $\left.\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ...   $(x^2-y^2,xy)$


$C_6$ $E$ $C_6$ $C_3$ $C_2$ ${C_3}^2$ ${C_6}^5$   $\varepsilon=\mathop{\rm exp}\nolimits (2\pi i/6)$
$A$ 1 1 1 1 1 1 $z, R_z$ $x^2+y^2, z^2$
$B$ 1 $-1$ 1 $-1$ 1 $-1$    
$E_1$ $\cases{\hfill 1\cr \hfill 1\cr}$ $\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $\left.\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $(yz,xz)$
$E_2$ $\cases{\hfill 1\cr \hfill 1\cr}$ $\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ... $\left.\vcenter{\normalbaselines\mathsurround=0pt
\ialign{$ ...   $(x^2-y^2,xy)$


$D_2$ $E$ $C_2(z)$ $C_2(y)$ $C_2(x)$    
$A_1$ 1 1 1 1   $x^2+y^2, z^2$
$B_1$ 1 1 $-1$ $-1$ $z, R_z$ $xy$
$B_2$ 1 $-1$ 1 $-1$ $y,R_y$ $xz$
$B_3$ 1 $-1$ $-1$ 1 $z, R_z$ $yz$


$D_3$ $E$ $2C_3$ $3C_2$    
$A_1$ 1 1 1   $x^2+y^2, z^2$
$A_2$ 1 1 $-1$ $z, R_z$ $xy$
$E$ 2 $-1$ 0 $(x,y)(R_x,R_y)$ $(x^2-y^2,xy)(xz,yz)$


$D_4$ $E$ $2C_4$ $C_2$ $2C_2'$ $2C_2''$    
$A_1$ 1 1 1 1 1   $x^2+y^2, z^2$
$A_2$ 1 1 1 $-1$ $-1$ $z, R_z$  
$B_1$ 1 $-1$ 1 1 $-1$   $x^2-y^2$
$B_2$ 1 $-1$ 1 $-1$ 1   $xy$
$E$ 2 0 $-2$ 0 0 $(x,y)(R_x,R_y)$ $(xz,yz)$


$D_5$ $E$ $2C_5$ $2{C_5}^2$ $5C_2$    
$A_1$ 1 1 1 1   $x^2+y^2, z^2$
$B_1$ 1 1 1 $-1$ $z, R_z$  
$B_2$ 2 $2\cos \phantom{0}72^\circ$ $2\cos 144^\circ$ 0 $(x,y)(R_x,R_y)$ $(xz,yz)$
$B_3$ 2 $2\cos 144^\circ$ $2\cos \phantom{0}72^\circ$ 0   $(x^2-y^2,xy)$


$D_6$ $E$ $2C_6$ $2C_3$ $C_2$ $3C_2'$ $3C_2''$    
$A_1$ 1 1 1 1 1 1   $x^2+y^2, z^2$
$A_2$ 1 1 1 1 $-1$ $-1$ $z, R_z$  
$B_1$ 1 $-1$ 1 $-1$ 1 $-1$    
$B_2$ 1 $-1$ 1 $-1$ $-1$ 1 $(x,y)(R_x,R_y)$  
$E_1$ 2 1 $-1$ $-2$ 0 0   $(xz,yz)$
$E_2$ 2 $-1$ $-1$ 2 0 0   $(x^2-y^2,xy)$


$C_{2v}$ $E$ $C_2$ $\sigma_v(xz)$ $\sigma_v'(yz)$    
$A_1$ 1 1 1 1 $z$ $x^2, y^2, z^2$
$A_2$ 1 1 $-1$ $-1$ $R_z$ $xy$
$B_1$ 1 $-1$ 1 $-1$ $x,R_y$ $xz$
$B_2$ 1 $-1$ $-1$ 1 $y,R_x$ $yz$


$C_{3v}$ $E$ $2C_3$ $3\sigma_v$    
$A_1$ 1 1 1 $z$ $x^2+y^2, z^2$
$A_2$ 1 1 $-1$ $R_z$  
$E$ 2 $-1$ 0 $(x,y)(R_x,R_y)$ $(x^2-y^2,xy)(xz,yz)$


$C_{4v}$ $E$ $2C_4$ $C_2$ $2\sigma_v$ $2\sigma_d$    
$A_1$ 1 1 1 1 1 $z$ $x^2+y^2, z^2$
$A_2$ 1 1 1 $-1$ $-1$ $R_z$  
$B_1$ 1 $-1$ 1 1 $-1$   $x^2-y^2$
$B_2$ 1 $-1$ 1 $-1$ 1   $xy$
$E$ 2 0 $-2$ 0 0 $(x,y)(R_x,R_y)$ $(xz,yz)$


$C_{5v}$ $E$ $2C_5$ $2{C_5}^2$ $5\sigma_v$    
$A_1$ 1 1 1 1 $z$ $x^2+y^2, z^2$
$B_1$ 1 1 1 $-1$ $R_z$  
$B_2$ 2 $2\cos \phantom{0}72^\circ$ $2\cos 144^\circ$ 0 $(x,y)(R_x,R_y)$ $(xz,yz)$
$B_3$ 2 $2\cos 144^\circ$ $2\cos \phantom{0}72^\circ$ 0   $(x^2-y^2,xy)$


$C_{6v}$ $E$ $2C_6$ $2C_3$ $C_2$ $3\sigma_v$ $3\sigma_d$    
$A_1$ 1 1 1 1 1 1 $z$ $x^2+y^2, z^2$
$A_2$ 1 1 1 1 $-1$ $-1$ $R_z$  
$B_1$ 1 $-1$ 1 $-1$ 1 $-1$    
$B_2$ 1 $-1$ 1 $-1$ $-1$ 1    
$E_1$ 2 1 $-1$ $-2$ 0 0 $(x,y)(R_x,R_y)$ $(xz,yz)$
$E_2$ 2 $-1$ $-1$ 2 0 0   $(x^2-y^2,xy)$


$C_{\infty v}$ $E$ ${C_\infty}^\Phi$ ... $\infty\sigma_v$    
$A_1\equiv\Sigma^+$ 1 1 ... 1 $z$ $x^2+y^2, z^2$
$A_2\equiv\Sigma^-$ 1 1 ... $-1$ $R_z$  
$E_1\equiv\Pi$ 2 $2\cos \phantom{0}\Phi$ ... 0 $(x,y); (R_x,R_y)$ $(xz,yz)$
$E_2\equiv\Delta$ 2 $2\cos 2\Phi$ ... 0   $(x^2-y^2,xy)$
$E_3\equiv\Phi$ 2 $2\cos 3\Phi$ ... 0    
$\vdots$ $\vdots$ $\vdots$ $\ddots$ $\vdots$    


References

Bishop, D. M. ``Character Tables.'' Appendix 1 in Group Theory and Chemistry. New York: Dover, pp. 279-288, 1993.

Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, 1990.

Iyanaga, S. and Kawada, Y. (Eds.). ``Characters of Finite Groups.'' Appendix B, Table 5 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1496-1503, 1980.



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© 1996-9 Eric W. Weisstein
1999-05-26