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    2181

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    Nonaxial Groups

    These groups are characterized by a lack of a proper rotation axis.

    \(C_1\) E
    A 1

     

    \(C_s\) E σh    
    A' 1 1 x, y, Rz x2, y2, z2, xy
    A" 1 -1 z, Rx, Ry yz, xz

     

    \(C_i\) E i    
    Ag 1 1 Rx, Ry, Rz x2, y2, z2, xy, yz, zx
    Au 1 -1 x,y,z  

    Cyclic \(C_n\) Groups

    These groups are characterized by an n-fold proper rotation axis \(C_n\).

    C2 E C2    
    A 1 1 z, Rz x2, y2, z2, xy
    B 1 -1 x, y, Rx, Ry yz,xz

     

    C3 E C3 C32   ε=exp(2π/3)
    A 1 1 1 z, Rz x2+y2, z2
    E \(\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\left. \begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix} \right\}\) (x,y), (Rx,Ry) (x2-y2, xy), (xz, yz)

     

    C4 E C4 C2 C43    
    A 1 1 1 1 z, Rz x2+y2, z2
    B 1 -1 1 -1   x2-y2, xy
    E \(\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.\) \(\begin{matrix} \;i \\ -i\end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\left. \begin{matrix} -i \\ \;i \end{matrix} \right\}\) (x,y), (Rx,Ry) (xz, yz)

     

    C5 E C5 C52 C53 C54   ε=exp(i2π/5)
    A 1 1 1 1 1 Z, Rz x2+y2, z2
    E1 \(\left\{ \begin{matrix}\sf 1 \\ 1 \end{matrix} \right.\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} \epsilon^2\; \\ \epsilon^{*2} \end{matrix}\) \(\begin{matrix} \epsilon^{*2} \\ \epsilon^2\; \end{matrix}\) \(\left. \begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix} \right\}\) (x, y), (Rx,Ry) (xz, yz)
    E2 \(\left\{ \begin{matrix}\sf 1 \\ 1 \end{matrix} \right.\) \(\begin{matrix} \epsilon^2\; \\ \epsilon^{*2} \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\left. \begin{matrix} \epsilon^{*2} \\ \epsilon^2\; \end{matrix} \right\}\)   (x2-y2, xy)

     

    C6 E C6 C3 C2 C32 C65   ε=exp(i2π/6)
    A 1 1 1 1 1 1 z, Rz x2+y2, z2
    B 1 -1 1 -1 1 -1    
    E1 \(\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} -\epsilon^* \\ -\epsilon\;\end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\) \(\left. \begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix} \right\}\) (Rx,Ry), (x,y) (xz, yz)
    E2 \(\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.\) \(\begin{matrix} -\epsilon^*\; \\ -\epsilon\; \end{matrix}\) \(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\) \(\begin{matrix} 1 \\ 1 \end{matrix}\) \(\begin{matrix} -\epsilon^*\; \\ -\epsilon\; \end{matrix}\) \(\left. \begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix} \right\}\)   (x2-y2, xy)

     

    C7 E C7 C72 C73 C74 C75 C76   ε=exp(i2π/7)
    A 1 1 1 1 1 1 1 z, Rz x2+y2, z2
    E1 \(\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} \epsilon^2\; \\ \epsilon^{*2} \end{matrix}\) \(\begin{matrix} \epsilon^3\; \\ \epsilon^{*3} \end{matrix}\) \(\begin{matrix} \epsilon^{*3}\; \\ \epsilon^3\; \end{matrix}\) \(\begin{matrix} \epsilon^{*2}\; \\ \epsilon^2\; \end{matrix}\) \(\left. \begin{matrix} \epsilon^{*} \\ \epsilon\; \end{matrix} \right\}\) (Rx,Ry), (x,y) (xz, yz)
    E2 \(\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.\) \(\begin{matrix} \epsilon^2\; \\ \epsilon^{*2} \end{matrix}\) \(\begin{matrix} \epsilon^{*3}\; \\ \epsilon^3\; \end{matrix}\) \(\begin{matrix} \epsilon^{*} \\ \epsilon\; \end{matrix}\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} \epsilon^3\; \\ \epsilon^{*3} \end{matrix}\) \(\left. \begin{matrix} \epsilon^{*2}\; \\ \epsilon^2\; \end{matrix} \right\}\)   (x2-y2, xy)
    E3 \(\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.\) \(\begin{matrix} \epsilon^3\; \\ \epsilon^{*3} \end{matrix}\) \(\begin{matrix} \epsilon^{*} \\ \epsilon\; \end{matrix}\) \(\begin{matrix} \epsilon^2\; \\ \epsilon^{*2} \end{matrix}\) \(\begin{matrix} \epsilon^{*2}\; \\ \epsilon^2\; \end{matrix}\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\left. \begin{matrix} \epsilon^{*3}\; \\ \epsilon^3\; \end{matrix} \right\}\)    

     

    C8 E C8 C4 C83 C2 C85 C43 C87   ε=exp(i2π/8)
    A 1 1 1 1 1 1 1 1 z, Rz x2+y2, z2
    B 1 -1 1 -1 1 -1 1 -1    
    E1 \(\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.\) \(\begin{matrix} \epsilon\; \\ \epsilon^*\end{matrix}\) \(\begin{matrix} \;i \\ -i \end{matrix}\) \(\begin{matrix} -\epsilon^* \\ -\epsilon\; \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\) \(\begin{matrix} -i \\ \;i \end{matrix}\) \(\left. \begin{matrix} \epsilon^{*} \\ \epsilon\; \end{matrix} \right\}\) (Rx,Ry), (x,y) (xz, yz)
    E2 \(\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.\) \(\begin{matrix} \;i \\ -i \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} -i \\ \;i \end{matrix}\) \(\begin{matrix} 1 \\ 1 \end{matrix}\) \(\begin{matrix} \;i \\ -i \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\left. \begin{matrix} -i \\ \;i \end{matrix} \right\}\)   (x2-y2, xy)
    E3 \(\left\{ \begin{matrix} 1 \\ 1 \end{matrix} \right.\) \(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\) \(\begin{matrix} \;i \\ -i\; \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} -i \\ \;i \end{matrix}\) \(\left. \begin{matrix} -\epsilon^* \\ -\epsilon\; \end{matrix} \right\}\)    

    Reflection \(C_{nh}\) Groups

    These groups are characterized by an n-fold proper rotation axis \(C_n\) and a mirror plane \(\sigma_h\) normal to \(C_n\).

    \(C_{2h}\) E C2 i σh    
    Ag 1 1 1 1 Rz x2, y2, z2
    Bg 1 -1 1 -1 Rx, Ry xz, yz
    Au 1 1 -1 -1 z  
    Bu 1 -1 -1 1 x,y  

     

    \(C_{3h}\) E C3 C32 σh S3 S35   ε=exp(i2π/3)
    A' 1 1 1 1 1 1 Rz x2+y2, z2
    E' \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\begin{matrix} 1 \\ 1 \end{matrix}\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\left.\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\right\}\) (x,y) (x2-y2, xy)
    A" 1 1 1 -1 -1 -1 z  
    E" \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\) \(\left.\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\right\}\) (Rx, Ry) (xz, yz)

     

    \(C_{4h}\) E C4 C2 C43 i S43 σh S4    
    Ag 1 1 1 1 1 1 1 1 Rz x2+y2, z2
    Bg 1 -1 1 -1 1 -1 1 -1   x2-y2, xy
    Eg \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} \; i \\ -i \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} -i \\ \; i \end{matrix}\) \(\begin{matrix} 1 \\ 1 \end{matrix}\) \(\begin{matrix} \; i \\ -i \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\left.\begin{matrix} -i \\ \; i \end{matrix}\right\}\) (Rx, Ry) (xz, yz)
    Au 1 1 1 1 -1 -1 -1 -1 z  
    Bu 1 -1 1 -1 -1 1 -1 1    
    Eu \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} \; i \\ -i \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} -i \\ \; i \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} -i \\ \; i \end{matrix}\) \(\begin{matrix} 1 \\ 1 \end{matrix}\) \(\left.\begin{matrix} \; i \\ -i \end{matrix}\right\}\) (x,y)  

     

    \(C_{5h}\) E C5 C52 C53 C54 σh S5 S57 S53 S59   ε=exp(i2π/5)
    A' 1 1 1 1 1 1 1 1 1 1 Rz x2+y2, z2
    E1' \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} \epsilon^{2}\; \\ \epsilon^{*2} \end{matrix}\) \(\begin{matrix} \epsilon^{*2} \\ \epsilon^2\; \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\begin{matrix} 1 \\ 1 \end{matrix}\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} \epsilon^{2}\; \\ \epsilon^{*2} \end{matrix}\) \(\begin{matrix} \epsilon^{*2} \\ \epsilon^2\; \end{matrix}\) \(\left.\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\right\}\) (x, y)  
    E2' \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} \epsilon^{2}\; \\ \epsilon^{*2} \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} \epsilon^{*2} \\ \epsilon^2\; \end{matrix}\) \(\begin{matrix} 1 \\ 1 \end{matrix}\) \(\begin{matrix} \epsilon^{2}\; \\ \epsilon^{*2} \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\left.\begin{matrix} \epsilon^{*2} \\ \epsilon^2\; \end{matrix}\right\}\)   (x2-y2, xy)
    A" 1 1 1 1 1 -1 -1 -1 -1 -1 z  
    E1" \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} \epsilon^{2}\; \\ \epsilon^{*2} \end{matrix}\) \(\begin{matrix} \epsilon^{*2} \\ \epsilon^2\; \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\) \(\begin{matrix} -\epsilon^{2}\; \\ -\epsilon^{*2} \end{matrix}\) \(\begin{matrix} -\epsilon^{*2} \\ -\epsilon^{2}\; \end{matrix}\) \(\left.\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\right\}\) (Rx, Ry) (xz, yz)
    E2" \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} \epsilon^{2}\; \\ \epsilon^{*2} \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} \epsilon^{*2} \\ \epsilon^2\; \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} -\epsilon^{2}\; \\ -\epsilon^{*2} \end{matrix}\) \(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\) \(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\) \(\left.\begin{matrix} -\epsilon^{*2} \\ -\epsilon^{2}\; \end{matrix}\right\}\)    

     

    \(C_{6h}\) E C6 C3 C2 C32 C65 i S35 S65 σh S6 S3   ε=exp(i2π/6)
    Ag 1 1 1 1 1 1 1 1 1 1 1 1 Rz x2+y2, z2
    Bg 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1    
    E1g \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\begin{matrix} 1 \\ 1 \end{matrix}\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\) \(\left.\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\right\}\) (Rx, Ry) (xz, yz)
    E2g \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\) \(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\) \(\begin{matrix} 1 \\ 1 \end{matrix}\) \(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\) \(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\) \(\begin{matrix} 1 \\ 1 \end{matrix}\) \(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\) \(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\) \(\begin{matrix} 1 \\ 1 \end{matrix}\) \(\begin{matrix} -\epsilon^* \\ -\epsilon\; \end{matrix}\) \(\left.\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\right\}\)   (x2-y2, xy)
    Au 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 z  
    Bu 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1    
    E1u \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\begin{matrix} 1 \\ 1 \end{matrix}\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\left.\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\right\}\) (x, y)  
    E2u \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\) \(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\) \(\begin{matrix} 1 \\ 1 \end{matrix}\) \(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\) \(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\left.\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\right\}\)    

    Pyramidal \(C_{nv}\) Groups

    These groups are characterized by an n-fold proper rotation axis \(C_n\) and n mirror planes \(σ_v\) which contain \(C_n\)

    \(C_{2v}\) E C2 σ(xz) σ(yz)    
    A1 1 1 1 1 z x2, y2, z2
    A2 1 1 -1 -1 Rz xy
    B1 1 -1 1 -1 x, Ry xz
    B2 1 -1 -1 1 y, Rx yz

     

    \(C_{3v}\) E 2C3 3σv    
    A1 1 1 1 z x2+y2, z2
    A2 1 1 -1 Rz  
    E 2 -1 0 (Rx, Ry), (x,y) (xz, yz) (x2-y2, xy)

     

    \(C_{4v}\) E 2C4 C2 2σv 2σd    
    A1 1 1 1 1 1 z x2+y2, z2
    A2 1 1 1 -1 -1 Rz  
    B1 1 -1 1 1 -1   x2-y2
    B2 1 -1 1 -1 1   xy
    E 2 0 -2 0 0 (Rx, Ry), (x,y) (xz, yz)

     

    \(C_{5v}\) E 2C5 2C52 5σv    
    A1 1 1 1 1 z x2+y2, z2
    A2 1 1 1 -1 Rz  
    E1 2 \(2\cos 72^\circ\) \(2\cos 144^\circ\) 0 (Rx, Ry), (x,y) (xz, yz)
    E2 2 \(2\cos{144^\circ}\) \(2\cos 72^\circ\) 0   (x2-y2, xy)

     

    \(C_{6v}\) E 2C6 2C3 C2 3σv 3σd    
    A1 1 1 1 1 1 1 z x2+y2, z2
    A2 1 1 1 1 -1 -1 Rz  
    B1 1 -1 1 -1 1 -1    
    B2 1 -1 1 -1 -1 1    
    E1 2 1 -1 -2 0 0 (Rx, Ry), (x,y) (xz, yz)
    E2 2 -1 -1 2 0 0   (x2-y2, xy)

     

    C∞v E 2C∞ ... ∞σv    
    A1 1 1 ... 1 z x2+y2, z2
    A2 1 1 ... -1 Rz  
    E1 2 \(2\cos{\phi}\) ... 0 (x,y), (Rx, Ry) (xz, yz)
    E2 2 \(2\cos{2\phi}\) ... 0   (x2-y2, xy)
    E3 2 \(2\cos{3\phi}\) ... 0    
    ... ... ... ... ...    

    Dihedral \(D_n\) Groups

    \(D_2\) E C2(z) C2(y) C2(x)    
    A 1 1 1 1   x2, y2, z2
    B1 1 1 -1 -1 z, Rz xy
    B2 1 -1 1 -1 y, Ry zx
    B3 1 -1 -1 1 x, Rx yz

     

    \(D_3\) E 2C3 3C2    
    A1 1 1 1   x2+y2, z2
    A2 1 1 -1 z, Rz  
    E 2 -1 0 (Rx, Ry), (x,y) (x2-y2, xy) (xz, yz)

     

    \(D_4\) E 2C4 C2(C42) 2C2' 2C2"    
    A1 1 1 1 1 1   x2+y2, z2
    A2 1 1 1 -1 -1 z, Rz  
    B1 1 -1 1 1 -1   x2-y2
    B2 1 -1 1 -1 1   xy
    E 2 0 -2 0 0 (Rx, Ry), (x,y) (xz, yz)

     

    2C52

    \(D_5\) E 2C5   5C2    
    A1 1 1 1 1   x2+y2, z2
    A2 1 1 1 -1 z, Rz  
    E1 2 \(2\cos{72^\circ}\) \(2\cos{144^\circ}\)   (Rx, Ry), (x,y) (xz, yz)
    E2 2 \(2\cos{144^\circ}\) \(2\cos{72^\circ}\)     (x2-y2, xy)

     

    \(D_6\) E 2C6 2C3 C2 2C2' 3C2"    
    A1 1 1 1 1 1 1   x2+y2, z2
    A2 1 1 1 1 -1 -1 z, Rz  
    B1 1 -1 1 -1 1 -1    
    B2 1 -1 1 -1 -1 1    
    E1 2 1 -1 -2 0 0 (Rx, Ry), (x,y) (xz, yz)
    E2 2 -1 -1 2 0 0   (x2-y2, xy)

    Prismatic \(D_{nh}\) Groups

    These groups are characterized by

    1. an n-fold proper rotation axis \(C_n\)
    2. n 2-fold proper rotation axes \(C_2\) normal to \(C_n\)
    3. a mirror plane \(\sigma_h\) normal to \(C_n\) and containing the \(C_2\) axes.
    \(D_{2h}\) E C2(z) C2(y) C2(x) i σ(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  

     

    \(D_{3h}\) E 2C3 3C2 σh 2S3 3σv    
    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)

     

    \(D_{4h}\) E 2C4 C2 2C2' 2C2" i 2S4 σh 2σv σd    
    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)  

     

    \(D_{5h}\) E 2C5 2C52 5C2 σh 2S5 2S53 5σv    
    A1' 1 1 1 1 1 1 1 1   x2+y2, z2
    A2' 1 1 1 -1 1 1 1 -1 Rz  
    E1' 2 \(2\cos{72^\circ}\) \(2\cos{144^\circ}\) 0 2 \(2\cos{72^\circ}\) \(2\cos{144^\circ}\)   (x,y)  
    E2' 2 \(2\cos{144^\circ}\) \(2\cos{72^\circ}\) 0 2 \(2\cos{144^\circ}\) \(2\cos{72^\circ}\)     (x2-y2, xy)
    A1" 1 1 1 1 -1 -1 -1 -1    
    A2" 1 1 1 -1 -1 -1 -1 1 z  
    E1" 2 \(2\cos{72^\circ}\) \(2\cos{144^\circ}\) 0 -2 \(-2\cos{72^\circ}\) \(-2\cos{144^\circ}\) 0 (Rx, Ry) (xz, yz)
    E2" 2 \(2\cos{144^\circ}\) \(2\cos{72^\circ}\) 0 -2 \(-2\cos{144^\circ}\) \(-2\cos{72^\circ}\) 0    

     

    D6h E 2C6 2C3 C2 3C2' 3C2" i 2S3 2S6 σh 3σd 3σv    
    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    

     

    D∞h E 2C∞ ... ∞σv i 2S∞ ... ∞ C2    
    Sg+ 1 1 ... 1 1 1 ... 1   x2+y2, z2
    Sg- 1 1 ... -1 1 1 ... -1 Rz  
    πg 2 \(2\cos{\phi}\) ... 0 2 \(-2\cos{\phi}\) ... 0 (Rx, Ry) (xz, yz)
    Dg 2 \(2\cos{2\phi}\) ... 0 2 \(2\cos{2\phi}\) ... 0   (x2-y2, xy)
    ... ... ... ... ...... ... ... ... ...    
    Su+ 1 1 ... 1 -1 -1 ... -1 z  
    Su- 1 1 ... -1 -1 -1 ... 1    
    πu 2 \(2\cos{\phi}\) ... 0 -2 \(2\cos{\phi}\) ... 0 (x, y)  
    Du 2 \(2\cos{2\phi}\) ... 0 -2 \(-2\cos{2\phi}\) ... 0    
    ... ... ... ... ... ... ... ... ...    

    Antiprismatic \(D_{nd}\) Groups

    These groups are characterized by

    1. an n-fold proper rotation axis Cn
    2. n 2-fold proper rotation axes C2 normal to Cn
    3. n mirror planes σd which contain Cn.
    D2d E 2S4 C2 2C2' 2σd    
    A1 1 1 1 1 1   x2+y2, z2
    A2 1 1 1 -1 -1 Rz  
    B1 1 -1 1 1 -1   x2-y2
    B2 1 -1 1 -1 1 z xy
    E 2 0 -2 0 0 (x, y), (Rx, Ry) (xz, yz)

     

    D3d E 2C3 3C2 i 2S6 3σd    
    A1g 1 1 1 1 1 1   x2+y2, z2
    A2g 1 1 -1 1 1 -1 Rz  
    Eg 2 -1 0 2 -1 0 (Rx, Ry) (x2-y2, xy),(xz, yz)
    A1u 1 1 1 -1 -1 -1    
    A2u 1 1 -1 -1 -1 1 z  
    Eu 2 -1 0 -2 1 0 (x, y)  

     

    D4d E 2S8 2C4 2S83 C2 4C2' 4σd    
    A1 1 1 1 1 1 1 1   x2+y2, z2
    A2 1 1 1 1 1 -1 -1 Rz  
    B1 1 -1 1 -1 1 1 -1    
    B2 1 -1 1 -1 1 -1 1 z  
    E1 2 \(\sqrt{2}\) 0 \(-\sqrt{2}\) -2 0 0 (x, y)  
    E2 2 0 -2 0 2 0 0   (x2-y2, xy)
    E3 2 \(-\sqrt{2}\) 0 \(\sqrt{2}\) -2 0 0 (Rx, Ry) (xz, yz)

     

    D5d E 2C5 2C52 5C2 i 2S103 2S10 5σd    
    A1g 1 1 1 1 1 1 1 1   x2+y2, z2
    A2g 1 1 1 -1 1 1 1 -1 Rz  
    E1g 2 \(2\cos 72^\circ\) \(2\cos 144^\circ\) 0 2 \(2\cos 72^\circ\) \(2\cos 144^\circ\) 0 (Rx, Ry) (xz, yz)
    E2g 2 \(2\cos 144^\circ\) \(2\cos 72^\circ\) 0 2 \(2\cos 144^\circ\) \(2\cos 72^\circ\) 0   (x2-y2, xy)
    A1u 1 1 1 1 -1 -1 -1 -1    
    A2u 1 1 1 -1 -1 1 -1 1 z  
    E1u 2 \(2\cos 72^\circ\) \(2\cos 144^\circ\) 0 -2 \(-2\cos 72^\circ\) \(-2\cos 144^\circ\) 0 (x, y)  
    E2u 2 \(2\cos 144^\circ\) \(2\cos 72^\circ\) 0 -2 \(-2\cos 144^\circ\) \(-2\cos 72^\circ\) 0    

     

    D6d E 2S12 2C6 2S4 2C3 2S125 C2 6C2' 6σd    
    A1 1 1 1 1 1 1 1 1 1   x2+y2, z2
    A2 1 1 1 1 1 1 1 -1 -1 Rz  
    B1 1 -1 1 -1 1 -1 1 1 -1    
    B2 1 -1 1 -1 1 -1 1 -1 1 z  
    E1 2 \(\sqrt{3}\) 1 0 -1 \(-\sqrt{3}\) -2 0 0 (x, y)  
    E2 2 1 -1 -2 -1 1 2 0 0   (x2-y2, xy)
    E3 2 0 -2 0 2 0 -2 0 0    
    E4 2 -1 -1 2 -1 -1 2 0 0    
    E5 2 \(-\sqrt{3}\) 1 0 -1 \(\sqrt{3}\) -2 0 0 (Rx, Ry) (xz, yz)

    Improper Rotation \(S_n\) Groups

    These groups are characterized by an n-fold improper rotation axis \(S_n\), where \(n\) is necessarily even

    \(S_4\) E S4 C2 S43    
    A 1 1 1 1 Rz x2+y2, z2
    B 1 -1 1 -1 z x2-y2, xy
    E \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} \; i \\ -i \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\left.\begin{matrix} -i \\ \; i \end{matrix}\right\}\) (x, y); (Rx, Ry) (xz, yz)

     

    S6 E C3 C32 i S65 S6    
    Ag 1 1 1 1 1 1 Rz x2+y2, z2
    Eg \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\begin{matrix} 1 \\ 1 \end{matrix}\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\left.\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\right\}\) (Rx, Ry) (x2-y2, xy), (xz, yz)
    Au 1 1 1 -1 -1 -1 z  
    Eu \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\) \(\left.\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\right\}\) (x, y)  

     

    S8 E S8 C4 S83 C2 S85 C43 S87   ε=exp(i2π/8)
    A 1 1 1 1 1 1 1 1 Rz x2+y2, z2
    B 1 -1 1 -1 1 -1 1 -1 z  
    E1 \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} \; i \\ -i \end{matrix}\) \(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\) \(\begin{matrix} -i \\ \; i \end{matrix}\) \(\left.\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\right\}\) (Rx, Ry), (x, y)  
    E2 \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} \; i \\ -i \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} -i \\ \; i \end{matrix}\) \(\begin{matrix} 1 \\ 1 \end{matrix}\) \(\begin{matrix} \; i \\ -i \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\left.\begin{matrix} -i \\ \; i \end{matrix}\right\}\)   (x2-y2, xy)
    E3 \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\) \(\begin{matrix} -i \\ \; i \end{matrix}\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\begin{matrix} \; i \\ -i \end{matrix}\) \(\left.\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\right\}\)   (xz, yz)

    Cubic Groups

    These polyhedral groups are characterized by not having a \(C_5\) proper rotation axis.

    \(T\) E 4C3 4C32 3C2    
    A 1 1 1 1   x2+y2+z2
    E \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\left.\begin{matrix} 1 \\ 1 \end{matrix}\right\}\)   (2z2-x2-y2, x2-y2)
    T 3 0 0   (Rx, Ry, Rz), (x, y, z) (xz, yz, xy)

     

    Th E 4C3 4C32 3C2 i 4S6 4S65 3σh   ε=exp(i2π/3)
    Ag 1 1 1 1 1 1 1 1   x2+y2+z2
    Eg \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\begin{matrix} 1 \\ 1 \end{matrix}\) \(\begin{matrix} 1 \\ 1 \end{matrix}\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\left.\begin{matrix} 1 \\ 1 \end{matrix}\right\}\)   (2z2-x2-y2, x2-y2)
    Tg 3 0 0 -1 1 0 0 -1 (Rx, Ry, Rz) (xz, yz, xy)
    Au 1 1 1 1 -1 -1 -1 -1    
    Eu \(\left\{\begin{matrix} 1 \\ 1 \end{matrix}\right.\) \(\begin{matrix} \epsilon\; \\ \epsilon^* \end{matrix}\) \(\begin{matrix} \epsilon^* \\ \epsilon\; \end{matrix}\) \(\begin{matrix} 1 \\ 1 \end{matrix}\) \(\begin{matrix} -1 \\ -1 \end{matrix}\) \(\begin{matrix} -\epsilon\; \\ -\epsilon^* \end{matrix}\) \(\begin{matrix} -\epsilon^{*} \\ -\epsilon\; \end{matrix}\) \(\left.\begin{matrix} -1 \\ -1 \end{matrix}\right\}\)    
    Tu 3 0 0 -1 -1 0 0 1 (x, y, z)  

     

    Td E 8C3 3C2 6S4 6σd    
    A1 1 1 1 1 1   x2+y2+z2
    A2 1 1 1 -1 -1    
    E 2 -1 2 0 0   (2z2-x2-y2, x2-y2)
    T1 3 0 -1 1 -1 (Rx, Ry, Rz)  
    T2 3 0 -1 -1 1 (x, y, z) (xz, yz, xy)

     

    O E 8C3 3C2 6C4 6C2    
    A1 1 1 1 1 1   x2+y2+z2
    A2 1 1 1 -1 -1    
    E 2 -1 2 0 0   (2z2-x2-y2, x2-y2)
    T1 3 0 -1 1 -1 (Rx, Ry, Rz), (x, y, z)  
    T2 3 0 -1 -1 1   (xz, yz, xy)

     

    Oh E 8C2 6C2 6C4 3C2(C42) i 6S4 8S6 3σh 6σd    
    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    
    Eg 2 -1 0 0 2 2 0 -1 2 0   (2z2-x2-y2, x2-y2)
    T1g 3 0 -1 1 -1 3 1 0 -1 -1 (Rx, Ry, Rz)  
    T2g 3 0 1 -1 -1 3 -1 0 -1 1   (xz, yz, xy)
    A1u 1 1 1 1 1 -1 -1 -1 -1 -1    
    A2u 1 1 -1 -1 1 -1 1 -1 -1 1    
    Eu 2 -1 0 0 2 -2 0 1 -2 0    
    T1u 3 0 -1 1 -1 -3 -1 0 1 1 (x, y, z)  
    T2u 3 0 1 -1 -1 -3 1 0 1 -1    

    Character Tables is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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