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$D_{n\mathrm{h}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccccccc|cc} \hline D_\mathrm{2h} & E & C_2(z) & C_2(y) & C_2(x) & i & \sigma(xy) & \sigma(xz) & \sigma(yz) & & \\ \hline \mathrm{A_g} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2,y^2,z^2 \\ \mathrm{B_{1g}} & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & R_y & xz \\ \mathrm{B_{3g}} & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & R_x & yz \\ \mathrm{A_u} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{B_{1u}} & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & y & \\ \mathrm{B_{3u}} & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & x & \\ \hline \end{array}$$

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$$\begin{array}{c|cccccc|cc} \hline D_\mathrm{3h} & E & 2C_3 & 3C_2 & \sigma_\mathrm{h} & 2S_3 & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_2'} & 1 & 1 & -1 & 1 & 1 & -1 & R_z & \\ \mathrm{E'} & 2 & -1 & 0 & 2 & -1 & 0 & (x,y) & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E''} & 2 & -1 & 0 & -2 & 1 & 0 & (R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

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$$\begin{array}{c|cccccccccc|cc} \hline D_\mathrm{4h} & E & 2C_4 & C_2 & 2C_2' & 2C_2'' & i & 2S_4 & \sigma_\mathrm{h} & 2\sigma_\mathrm{v} & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_{1g}} & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E_g} & 2 & 0 & -2 & 0 & 0 & 2 & 0 & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{1u}} & 1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{E_u} & 2 & 0 & -2 & 0 & 0 & -2 & 0 & 2 & 0 & 0 & (x,y) & \\ \hline \end{array}$$

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$$\small \begin{array}{c|cccccccc|cc} \hline D_\mathrm{5h} & E & 2C_5 & 2C_5^2 & 5C_2 & \sigma_\mathrm{h} & 2S_5 & 2S_5^3 & 5\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2, z^2 \\ \mathrm{A_2'} & 1 & 1 & 1 & -1 & 1 & 1 & 1 & -1 & R_z & \\ \mathrm{E_1'} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & (x,y) & \\ \mathrm{E_2'} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E_1''} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & -2 & -2\cos 72^\circ & -2\cos 144^\circ & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_2''} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & -2 & -2\cos 144^\circ & -2\cos 72^\circ & 0 & & \\ \hline \end{array}$$

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$$\small \begin{array}{c|cccccccccccc|cc} \hline D_\mathrm{6h} & E & 2C_6 & 2C_3 & C_2 & 3C_2' & 3C_2'' & i & 2S_3 & 2S_6 & \sigma_\mathrm{h} & 3\sigma_\mathrm{d} & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_{1g}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & -1 & 1 & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_{1g}} & 2 & 1 & -1 & -2 & 0 & 0 & 2 & 1 & -1 & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_{2g}} & 2 & -1 & -1 & 2 & 0 & 0 & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{1u}} & 1 & -1 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & \\ \mathrm{E_{1u}} & 2 & 1 & -1 & -2 & 0 & 0 & -2 & -1 & 1 & 2 & 0 & 0 & (x,y) & \\ \mathrm{E_{2u}} & 2 & -1 & -1 & 2 & 0 & 0 & -2 & 1 & 1 & -2 & 0 & 0 & & \\ \hline \end{array}$$

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(back to top) $\,\,\,$

$D_{n\mathrm{h}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccccccc|cc} \hline D_\mathrm{2h} & E & C_2(z) & C_2(y) & C_2(x) & i & \sigma(xy) & \sigma(xz) & \sigma(yz) & & \\ \hline \mathrm{A_g} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2,y^2,z^2 \\ \mathrm{B_{1g}} & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & R_y & xz \\ \mathrm{B_{3g}} & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & R_x & yz \\ \mathrm{A_u} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{B_{1u}} & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & y & \\ \mathrm{B_{3u}} & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & x & \\ \hline \end{array}$$

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$$\begin{array}{c|cccccc|cc} \hline D_\mathrm{3h} & E & 2C_3 & 3C_2 & \sigma_\mathrm{h} & 2S_3 & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_2'} & 1 & 1 & -1 & 1 & 1 & -1 & R_z & \\ \mathrm{E'} & 2 & -1 & 0 & 2 & -1 & 0 & (x,y) & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E''} & 2 & -1 & 0 & -2 & 1 & 0 & (R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

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$$\begin{array}{c|cccccccccc|cc} \hline D_\mathrm{4h} & E & 2C_4 & C_2 & 2C_2' & 2C_2'' & i & 2S_4 & \sigma_\mathrm{h} & 2\sigma_\mathrm{v} & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_{1g}} & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E_g} & 2 & 0 & -2 & 0 & 0 & 2 & 0 & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{1u}} & 1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{E_u} & 2 & 0 & -2 & 0 & 0 & -2 & 0 & 2 & 0 & 0 & (x,y) & \\ \hline \end{array}$$

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$$\small \begin{array}{c|cccccccc|cc} \hline D_\mathrm{5h} & E & 2C_5 & 2C_5^2 & 5C_2 & \sigma_\mathrm{h} & 2S_5 & 2S_5^3 & 5\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2, z^2 \\ \mathrm{A_2'} & 1 & 1 & 1 & -1 & 1 & 1 & 1 & -1 & R_z & \\ \mathrm{E_1'} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & (x,y) & \\ \mathrm{E_2'} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E_1''} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & -2 & -2\cos 72^\circ & -2\cos 144^\circ & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_2''} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & -2 & -2\cos 144^\circ & -2\cos 72^\circ & 0 & & \\ \hline \end{array}$$

$\,$

$$\small \begin{array}{c|cccccccccccc|cc} \hline D_\mathrm{6h} & E & 2C_6 & 2C_3 & C_2 & 3C_2' & 3C_2'' & i & 2S_3 & 2S_6 & \sigma_\mathrm{h} & 3\sigma_\mathrm{d} & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_{1g}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & -1 & 1 & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_{1g}} & 2 & 1 & -1 & -2 & 0 & 0 & 2 & 1 & -1 & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_{2g}} & 2 & -1 & -1 & 2 & 0 & 0 & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{1u}} & 1 & -1 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & \\ \mathrm{E_{1u}} & 2 & 1 & -1 & -2 & 0 & 0 & -2 & -1 & 1 & 2 & 0 & 0 & (x,y) & \\ \mathrm{E_{2u}} & 2 & -1 & -1 & 2 & 0 & 0 & -2 & 1 & 1 & -2 & 0 & 0 & & \\ \hline \end{array}$$

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(back to top)

$D_{n\mathrm{h}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccccccc|cc} \hline D_\mathrm{2h} & E & C_2(z) & C_2(y) & C_2(x) & i & \sigma(xy) & \sigma(xz) & \sigma(yz) & & \\ \hline \mathrm{A_g} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2,y^2,z^2 \\ \mathrm{B_{1g}} & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & R_y & xz \\ \mathrm{B_{3g}} & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & R_x & yz \\ \mathrm{A_u} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{B_{1u}} & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & y & \\ \mathrm{B_{3u}} & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & x & \\ \hline \end{array}$$

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$$\begin{array}{c|cccccc|cc} \hline D_\mathrm{3h} & E & 2C_3 & 3C_2 & \sigma_\mathrm{h} & 2S_3 & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_2'} & 1 & 1 & -1 & 1 & 1 & -1 & R_z & \\ \mathrm{E'} & 2 & -1 & 0 & 2 & -1 & 0 & (x,y) & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E''} & 2 & -1 & 0 & -2 & 1 & 0 & (R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

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$$\begin{array}{c|cccccccccc|cc} \hline D_\mathrm{4h} & E & 2C_4 & C_2 & 2C_2' & 2C_2'' & i & 2S_4 & \sigma_\mathrm{h} & 2\sigma_\mathrm{v} & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_{1g}} & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E_g} & 2 & 0 & -2 & 0 & 0 & 2 & 0 & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{1u}} & 1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{E_u} & 2 & 0 & -2 & 0 & 0 & -2 & 0 & 2 & 0 & 0 & (x,y) & \\ \hline \end{array}$$

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$$\small \begin{array}{c|cccccccc|cc} \hline D_\mathrm{5h} & E & 2C_5 & 2C_5^2 & 5C_2 & \sigma_\mathrm{h} & 2S_5 & 2S_5^3 & 5\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2, z^2 \\ \mathrm{A_2'} & 1 & 1 & 1 & -1 & 1 & 1 & 1 & -1 & R_z & \\ \mathrm{E_1'} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & (x,y) & \\ \mathrm{E_2'} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E_1''} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & -2 & -2\cos 72^\circ & -2\cos 144^\circ & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_2''} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & -2 & -2\cos 144^\circ & -2\cos 72^\circ & 0 & & \\ \hline \end{array}$$

$\,$

$$\small \begin{array}{c|cccccccccccc|cc} \hline D_\mathrm{6h} & E & 2C_6 & 2C_3 & C_2 & 3C_2' & 3C_2'' & i & 2S_3 & 2S_6 & \sigma_\mathrm{h} & 3\sigma_\mathrm{d} & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_{1g}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & -1 & 1 & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_{1g}} & 2 & 1 & -1 & -2 & 0 & 0 & 2 & 1 & -1 & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_{2g}} & 2 & -1 & -1 & 2 & 0 & 0 & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{1u}} & 1 & -1 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & \\ \mathrm{E_{1u}} & 2 & 1 & -1 & -2 & 0 & 0 & -2 & -1 & 1 & 2 & 0 & 0 & (x,y) & \\ \mathrm{E_{2u}} & 2 & -1 & -1 & 2 & 0 & 0 & -2 & 1 & 1 & -2 & 0 & 0 & & \\ \hline \end{array}$$

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$D_{n\mathrm{h}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccccccc|cc} \hline D_\mathrm{2h} & E & C_2(z) & C_2(y) & C_2(x) & i & \sigma(xy) & \sigma(xz) & \sigma(yz) & & \\ \hline \mathrm{A_g} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2,y^2,z^2 \\ \mathrm{B_{1g}} & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & R_y & xz \\ \mathrm{B_{3g}} & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & R_x & yz \\ \mathrm{A_u} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{B_{1u}} & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & y & \\ \mathrm{B_{3u}} & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & x & \\ \hline \end{array}$$

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$$\begin{array}{c|cccccc|cc} \hline D_\mathrm{3h} & E & 2C_3 & 3C_2 & \sigma_\mathrm{h} & 2S_3 & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_2'} & 1 & 1 & -1 & 1 & 1 & -1 & R_z & \\ \mathrm{E'} & 2 & -1 & 0 & 2 & -1 & 0 & (x,y) & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E''} & 2 & -1 & 0 & -2 & 1 & 0 & (R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

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$$\begin{array}{c|cccccccccc|cc} \hline D_\mathrm{4h} & E & 2C_4 & C_2 & 2C_2' & 2C_2'' & i & 2S_4 & \sigma_\mathrm{h} & 2\sigma_\mathrm{v} & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_{1g}} & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E_g} & 2 & 0 & -2 & 0 & 0 & 2 & 0 & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{1u}} & 1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{E_u} & 2 & 0 & -2 & 0 & 0 & -2 & 0 & 2 & 0 & 0 & (x,y) & \\ \hline \end{array}$$

$\,$

$$\small \begin{array}{c|cccccccc|cc} \hline D_\mathrm{5h} & E & 2C_5 & 2C_5^2 & 5C_2 & \sigma_\mathrm{h} & 2S_5 & 2S_5^3 & 5\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2, z^2 \\ \mathrm{A_2'} & 1 & 1 & 1 & -1 & 1 & 1 & 1 & -1 & R_z & \\ \mathrm{E_1'} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & (x,y) & \\ \mathrm{E_2'} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E_1''} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & -2 & -2\cos 72^\circ & -2\cos 144^\circ & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_2''} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & -2 & -2\cos 144^\circ & -2\cos 72^\circ & 0 & & \\ \hline \end{array}$$

$\,$

$$\small \begin{array}{c|cccccccccccc|cc} \hline D_\mathrm{6h} & E & 2C_6 & 2C_3 & C_2 & 3C_2' & 3C_2'' & i & 2S_3 & 2S_6 & \sigma_\mathrm{h} & 3\sigma_\mathrm{d} & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_{1g}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & -1 & 1 & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_{1g}} & 2 & 1 & -1 & -2 & 0 & 0 & 2 & 1 & -1 & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_{2g}} & 2 & -1 & -1 & 2 & 0 & 0 & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{1u}} & 1 & -1 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & \\ \mathrm{E_{1u}} & 2 & 1 & -1 & -2 & 0 & 0 & -2 & -1 & 1 & 2 & 0 & 0 & (x,y) & \\ \mathrm{E_{2u}} & 2 & -1 & -1 & 2 & 0 & 0 & -2 & 1 & 1 & -2 & 0 & 0 & & \\ \hline \end{array}$$

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$D_{n\mathrm{h}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccccccc|cc} \hline D_\mathrm{2h} & E & C_2(z) & C_2(y) & C_2(x) & i & \sigma(xy) & \sigma(xz) & \sigma(yz) & & \\ \hline \mathrm{A_g} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2,y^2,z^2 \\ \mathrm{B_{1g}} & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & R_y & xz \\ \mathrm{B_{3g}} & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & R_x & yz \\ \mathrm{A_u} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{B_{1u}} & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & y & \\ \mathrm{B_{3u}} & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & x & \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccccc|cc} \hline D_\mathrm{3h} & E & 2C_3 & 3C_2 & \sigma_\mathrm{h} & 2S_3 & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_2'} & 1 & 1 & -1 & 1 & 1 & -1 & R_z & \\ \mathrm{E'} & 2 & -1 & 0 & 2 & -1 & 0 & (x,y) & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E''} & 2 & -1 & 0 & -2 & 1 & 0 & (R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccccccccc|cc} \hline D_\mathrm{4h} & E & 2C_4 & C_2 & 2C_2' & 2C_2'' & i & 2S_4 & \sigma_\mathrm{h} & 2\sigma_\mathrm{v} & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_{1g}} & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E_g} & 2 & 0 & -2 & 0 & 0 & 2 & 0 & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{1u}} & 1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{E_u} & 2 & 0 & -2 & 0 & 0 & -2 & 0 & 2 & 0 & 0 & (x,y) & \\ \hline \end{array}$$

$\,$

$$\small \begin{array}{c|cccccccc|cc} \hline D_\mathrm{5h} & E & 2C_5 & 2C_5^2 & 5C_2 & \sigma_\mathrm{h} & 2S_5 & 2S_5^3 & 5\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2, z^2 \\ \mathrm{A_2'} & 1 & 1 & 1 & -1 & 1 & 1 & 1 & -1 & R_z & \\ \mathrm{E_1'} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & (x,y) & \\ \mathrm{E_2'} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E_1''} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & -2 & -2\cos 72^\circ & -2\cos 144^\circ & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_2''} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & -2 & -2\cos 144^\circ & -2\cos 72^\circ & 0 & & \\ \hline \end{array}$$

$\,$

$$\small \begin{array}{c|cccccccccccc|cc} \hline D_\mathrm{6h} & E & 2C_6 & 2C_3 & C_2 & 3C_2' & 3C_2'' & i & 2S_3 & 2S_6 & \sigma_\mathrm{h} & 3\sigma_\mathrm{d} & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_{1g}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & -1 & 1 & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_{1g}} & 2 & 1 & -1 & -2 & 0 & 0 & 2 & 1 & -1 & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_{2g}} & 2 & -1 & -1 & 2 & 0 & 0 & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{1u}} & 1 & -1 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & \\ \mathrm{E_{1u}} & 2 & 1 & -1 & -2 & 0 & 0 & -2 & -1 & 1 & 2 & 0 & 0 & (x,y) & \\ \mathrm{E_{2u}} & 2 & -1 & -1 & 2 & 0 & 0 & -2 & 1 & 1 & -2 & 0 & 0 & & \\ \hline \end{array}$$

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$D_{n\mathrm{h}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccccccc|cc} \hline D_\mathrm{2h} & E & C_2(z) & C_2(y) & C_2(x) & i & \sigma(xy) & \sigma(xz) & \sigma(yz) & & \\ \hline \mathrm{A_g} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2,y^2,z^2 \\ \mathrm{B_{1g}} & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & R_y & xz \\ \mathrm{B_{3g}} & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & R_x & yz \\ \mathrm{A_u} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{B_{1u}} & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & y & \\ \mathrm{B_{3u}} & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & x & \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccccc|cc} \hline D_\mathrm{3h} & E & 2C_3 & 3C_2 & \sigma_\mathrm{h} & 2S_3 & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_2'} & 1 & 1 & -1 & 1 & 1 & -1 & R_z & \\ \mathrm{E'} & 2 & -1 & 0 & 2 & -1 & 0 & (x,y) & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E''} & 2 & -1 & 0 & -2 & 1 & 0 & (R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccccccccc|cc} \hline D_\mathrm{4h} & E & 2C_4 & C_2 & 2C_2' & 2C_2'' & i & 2S_4 & \sigma_\mathrm{h} & 2\sigma_\mathrm{v} & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_{1g}} & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E_g} & 2 & 0 & -2 & 0 & 0 & 2 & 0 & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{1u}} & 1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{E_u} & 2 & 0 & -2 & 0 & 0 & -2 & 0 & 2 & 0 & 0 & (x,y) & \\ \hline \end{array}$$

$\,$

$$\small \begin{array}{c|cccccccc|cc} \hline D_\mathrm{5h} & E & 2C_5 & 2C_5^2 & 5C_2 & \sigma_\mathrm{h} & 2S_5 & 2S_5^3 & 5\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2, z^2 \\ \mathrm{A_2'} & 1 & 1 & 1 & -1 & 1 & 1 & 1 & -1 & R_z & \\ \mathrm{E_1'} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & (x,y) & \\ \mathrm{E_2'} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E_1''} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & -2 & -2\cos 72^\circ & -2\cos 144^\circ & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_2''} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & -2 & -2\cos 144^\circ & -2\cos 72^\circ & 0 & & \\ \hline \end{array}$$

$\,$

$$\small \begin{array}{c|cccccccccccc|cc} \hline D_\mathrm{6h} & E & 2C_6 & 2C_3 & C_2 & 3C_2' & 3C_2'' & i & 2S_3 & 2S_6 & \sigma_\mathrm{h} & 3\sigma_\mathrm{d} & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_{1g}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & -1 & 1 & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_{1g}} & 2 & 1 & -1 & -2 & 0 & 0 & 2 & 1 & -1 & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_{2g}} & 2 & -1 & -1 & 2 & 0 & 0 & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{1u}} & 1 & -1 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & \\ \mathrm{E_{1u}} & 2 & 1 & -1 & -2 & 0 & 0 & -2 & -1 & 1 & 2 & 0 & 0 & (x,y) & \\ \mathrm{E_{2u}} & 2 & -1 & -1 & 2 & 0 & 0 & -2 & 1 & 1 & -2 & 0 & 0 & & \\ \hline \end{array}$$

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$D_{n\mathrm{h}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccccccc|cc} \hline D_\mathrm{2h} & E & C_2(z) & C_2(y) & C_2(x) & i & \sigma(xy) & \sigma(xz) & \sigma(yz) & & \\ \hline \mathrm{A_g} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2,y^2,z^2 \\ \mathrm{B_{1g}} & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & R_y & xz \\ \mathrm{B_{3g}} & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & R_x & yz \\ \mathrm{A_u} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{B_{1u}} & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & y & \\ \mathrm{B_{3u}} & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & x & \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccccc|cc} \hline D_\mathrm{3h} & E & 2C_3 & 3C_2 & \sigma_\mathrm{h} & 2S_3 & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_2'} & 1 & 1 & -1 & 1 & 1 & -1 & R_z & \\ \mathrm{E'} & 2 & -1 & 0 & 2 & -1 & 0 & (x,y) & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E''} & 2 & -1 & 0 & -2 & 1 & 0 & (R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccccccccc|cc} \hline D_\mathrm{4h} & E & 2C_4 & C_2 & 2C_2' & 2C_2'' & i & 2S_4 & \sigma_\mathrm{h} & 2\sigma_\mathrm{v} & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_{1g}} & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E_g} & 2 & 0 & -2 & 0 & 0 & 2 & 0 & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{1u}} & 1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{E_u} & 2 & 0 & -2 & 0 & 0 & -2 & 0 & 2 & 0 & 0 & (x,y) & \\ \hline \end{array}$$

$\,$

$$\small \begin{array}{c|cccccccc|cc} \hline D_\mathrm{5h} & E & 2C_5 & 2C_5^2 & 5C_2 & \sigma_\mathrm{h} & 2S_5 & 2S_5^3 & 5\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2, z^2 \\ \mathrm{A_2'} & 1 & 1 & 1 & -1 & 1 & 1 & 1 & -1 & R_z & \\ \mathrm{E_1'} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & (x,y) & \\ \mathrm{E_2'} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E_1''} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & -2 & -2\cos 72^\circ & -2\cos 144^\circ & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_2''} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & -2 & -2\cos 144^\circ & -2\cos 72^\circ & 0 & & \\ \hline \end{array}$$

$\,$

$$\small \begin{array}{c|cccccccccccc|cc} \hline D_\mathrm{6h} & E & 2C_6 & 2C_3 & C_2 & 3C_2' & 3C_2'' & i & 2S_3 & 2S_6 & \sigma_\mathrm{h} & 3\sigma_\mathrm{d} & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_{1g}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & -1 & 1 & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_{1g}} & 2 & 1 & -1 & -2 & 0 & 0 & 2 & 1 & -1 & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_{2g}} & 2 & -1 & -1 & 2 & 0 & 0 & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{1u}} & 1 & -1 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & \\ \mathrm{E_{1u}} & 2 & 1 & -1 & -2 & 0 & 0 & -2 & -1 & 1 & 2 & 0 & 0 & (x,y) & \\ \mathrm{E_{2u}} & 2 & -1 & -1 & 2 & 0 & 0 & -2 & 1 & 1 & -2 & 0 & 0 & & \\ \hline \end{array}$$

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$D_{n\mathrm{h}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccccccc|cc} \hline D_\mathrm{2h} & E & C_2(z) & C_2(y) & C_2(x) & i & \sigma(xy) & \sigma(xz) & \sigma(yz) & & \\ \hline \mathrm{A_g} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2,y^2,z^2 \\ \mathrm{B_{1g}} & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & R_y & xz \\ \mathrm{B_{3g}} & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & R_x & yz \\ \mathrm{A_u} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{B_{1u}} & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & y & \\ \mathrm{B_{3u}} & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & x & \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccccc|cc} \hline D_\mathrm{3h} & E & 2C_3 & 3C_2 & \sigma_\mathrm{h} & 2S_3 & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_2'} & 1 & 1 & -1 & 1 & 1 & -1 & R_z & \\ \mathrm{E'} & 2 & -1 & 0 & 2 & -1 & 0 & (x,y) & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E''} & 2 & -1 & 0 & -2 & 1 & 0 & (R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccccccccc|cc} \hline D_\mathrm{4h} & E & 2C_4 & C_2 & 2C_2' & 2C_2'' & i & 2S_4 & \sigma_\mathrm{h} & 2\sigma_\mathrm{v} & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_{1g}} & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E_g} & 2 & 0 & -2 & 0 & 0 & 2 & 0 & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{1u}} & 1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{E_u} & 2 & 0 & -2 & 0 & 0 & -2 & 0 & 2 & 0 & 0 & (x,y) & \\ \hline \end{array}$$

$\,$

$$\small \begin{array}{c|cccccccc|cc} \hline D_\mathrm{5h} & E & 2C_5 & 2C_5^2 & 5C_2 & \sigma_\mathrm{h} & 2S_5 & 2S_5^3 & 5\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2, z^2 \\ \mathrm{A_2'} & 1 & 1 & 1 & -1 & 1 & 1 & 1 & -1 & R_z & \\ \mathrm{E_1'} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & (x,y) & \\ \mathrm{E_2'} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E_1''} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & -2 & -2\cos 72^\circ & -2\cos 144^\circ & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_2''} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & -2 & -2\cos 144^\circ & -2\cos 72^\circ & 0 & & \\ \hline \end{array}$$

$\,$

$$\small \begin{array}{c|cccccccccccc|cc} \hline D_\mathrm{6h} & E & 2C_6 & 2C_3 & C_2 & 3C_2' & 3C_2'' & i & 2S_3 & 2S_6 & \sigma_\mathrm{h} & 3\sigma_\mathrm{d} & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_{1g}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & -1 & 1 & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_{1g}} & 2 & 1 & -1 & -2 & 0 & 0 & 2 & 1 & -1 & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_{2g}} & 2 & -1 & -1 & 2 & 0 & 0 & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{1u}} & 1 & -1 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & \\ \mathrm{E_{1u}} & 2 & 1 & -1 & -2 & 0 & 0 & -2 & -1 & 1 & 2 & 0 & 0 & (x,y) & \\ \mathrm{E_{2u}} & 2 & -1 & -1 & 2 & 0 & 0 & -2 & 1 & 1 & -2 & 0 & 0 & & \\ \hline \end{array}$$

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$D_{n\mathrm{h}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccccccc|cc} \hline D_\mathrm{2h} & E & C_2(z) & C_2(y) & C_2(x) & i & \sigma(xy) & \sigma(xz) & \sigma(yz) & & \\ \hline \mathrm{A_g} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2,y^2,z^2 \\ \mathrm{B_{1g}} & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & R_y & xz \\ \mathrm{B_{3g}} & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & R_x & yz \\ \mathrm{A_u} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{B_{1u}} & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & y & \\ \mathrm{B_{3u}} & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & x & \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccccc|cc} \hline D_\mathrm{3h} & E & 2C_3 & 3C_2 & \sigma_\mathrm{h} & 2S_3 & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_2'} & 1 & 1 & -1 & 1 & 1 & -1 & R_z & \\ \mathrm{E'} & 2 & -1 & 0 & 2 & -1 & 0 & (x,y) & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E''} & 2 & -1 & 0 & -2 & 1 & 0 & (R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

$\,$

$$\begin{array}{c|cccccccccc|cc} \hline D_\mathrm{4h} & E & 2C_4 & C_2 & 2C_2' & 2C_2'' & i & 2S_4 & \sigma_\mathrm{h} & 2\sigma_\mathrm{v} & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_{1g}} & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E_g} & 2 & 0 & -2 & 0 & 0 & 2 & 0 & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{1u}} & 1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{E_u} & 2 & 0 & -2 & 0 & 0 & -2 & 0 & 2 & 0 & 0 & (x,y) & \\ \hline \end{array}$$

$\,$

$$\small \begin{array}{c|cccccccc|cc} \hline D_\mathrm{5h} & E & 2C_5 & 2C_5^2 & 5C_2 & \sigma_\mathrm{h} & 2S_5 & 2S_5^3 & 5\sigma_\mathrm{v} & & \\ \hline \mathrm{A_1'} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2, z^2 \\ \mathrm{A_2'} & 1 & 1 & 1 & -1 & 1 & 1 & 1 & -1 & R_z & \\ \mathrm{E_1'} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & (x,y) & \\ \mathrm{E_2'} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & & (x^2-y^2,xy) \\ \mathrm{A_1''} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_2''} & 1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & z & \\ \mathrm{E_1''} & 2 & 2\cos 72^\circ & 2\cos 144^\circ & 0 & -2 & -2\cos 72^\circ & -2\cos 144^\circ & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_2''} & 2 & 2\cos 144^\circ & 2\cos 72^\circ & 0 & -2 & -2\cos 144^\circ & -2\cos 72^\circ & 0 & & \\ \hline \end{array}$$

$\,$

$$\small \begin{array}{c|cccccccccccc|cc} \hline D_\mathrm{6h} & E & 2C_6 & 2C_3 & C_2 & 3C_2' & 3C_2'' & i & 2S_3 & 2S_6 & \sigma_\mathrm{h} & 3\sigma_\mathrm{d} & 3\sigma_\mathrm{v} & & \\ \hline \mathrm{A_{1g}} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & & x^2+y^2,z^2 \\ \mathrm{A_{2g}} & 1 & 1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_{1g}} & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_{2g}} & 1 & -1 & 1 & -1 & -1 & 1 & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_{1g}} & 2 & 1 & -1 & -2 & 0 & 0 & 2 & 1 & -1 & -2 & 0 & 0 & (R_x,R_y) & (xz,yz) \\ \mathrm{E_{2g}} & 2 & -1 & -1 & 2 & 0 & 0 & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \mathrm{A_{1u}} & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & & \\ \mathrm{A_{2u}} & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & z & \\ \mathrm{B_{1u}} & 1 & -1 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & & \\ \mathrm{B_{2u}} & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & & \\ \mathrm{E_{1u}} & 2 & 1 & -1 & -2 & 0 & 0 & -2 & -1 & 1 & 2 & 0 & 0 & (x,y) & \\ \mathrm{E_{2u}} & 2 & -1 & -1 & 2 & 0 & 0 & -2 & 1 & 1 & -2 & 0 & 0 & & \\ \hline \end{array}$$

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