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$C_{n\mathrm{v}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccc|cc} \hline C_\mathrm{2v} & E & C_2 & \sigma_\mathrm{v}(xz) & \sigma_\mathrm{v}'(yz) & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & z & x^2, y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & x, R_y & xz \\ \mathrm{B_2} & 1 & -1 & -1 & 1 & y, R_x & yz \\ \hline \end{array}$$

$\,$

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

$\,$

$$\begin{array}{c|ccccc|cc} \hline C_\mathrm{4v} & E & 2C_4 & C_2 & 2\sigma_\mathrm{v} & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E} & 2 & 0 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

$\,$

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

$\,$

$$\begin{array}{c|cccccc|cc} \hline C_\mathrm{6v} & E & 2C_6 & 2C_3 & C_2 & 3\sigma_\mathrm{v} & 3\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2,z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_1} & 2 & 1 & -1 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \mathrm{E_2} & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \hline \end{array}$$

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$C_{n\mathrm{v}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccc|cc} \hline C_\mathrm{2v} & E & C_2 & \sigma_\mathrm{v}(xz) & \sigma_\mathrm{v}'(yz) & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & z & x^2, y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & x, R_y & xz \\ \mathrm{B_2} & 1 & -1 & -1 & 1 & y, R_x & yz \\ \hline \end{array}$$

$\,$

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

$\,$

$$\begin{array}{c|ccccc|cc} \hline C_\mathrm{4v} & E & 2C_4 & C_2 & 2\sigma_\mathrm{v} & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E} & 2 & 0 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

$\,$

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

$\,$

$$\begin{array}{c|cccccc|cc} \hline C_\mathrm{6v} & E & 2C_6 & 2C_3 & C_2 & 3\sigma_\mathrm{v} & 3\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2,z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_1} & 2 & 1 & -1 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \mathrm{E_2} & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \hline \end{array}$$

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$C_{n\mathrm{v}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccc|cc} \hline C_\mathrm{2v} & E & C_2 & \sigma_\mathrm{v}(xz) & \sigma_\mathrm{v}'(yz) & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & z & x^2, y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & x, R_y & xz \\ \mathrm{B_2} & 1 & -1 & -1 & 1 & y, R_x & yz \\ \hline \end{array}$$

$\,$

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

$\,$

$$\begin{array}{c|ccccc|cc} \hline C_\mathrm{4v} & E & 2C_4 & C_2 & 2\sigma_\mathrm{v} & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E} & 2 & 0 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

$\,$

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

$\,$

$$\begin{array}{c|cccccc|cc} \hline C_\mathrm{6v} & E & 2C_6 & 2C_3 & C_2 & 3\sigma_\mathrm{v} & 3\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2,z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_1} & 2 & 1 & -1 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \mathrm{E_2} & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \hline \end{array}$$

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$C_{n\mathrm{v}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccc|cc} \hline C_\mathrm{2v} & E & C_2 & \sigma_\mathrm{v}(xz) & \sigma_\mathrm{v}'(yz) & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & z & x^2, y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & x, R_y & xz \\ \mathrm{B_2} & 1 & -1 & -1 & 1 & y, R_x & yz \\ \hline \end{array}$$

$\,$

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

$\,$

$$\begin{array}{c|ccccc|cc} \hline C_\mathrm{4v} & E & 2C_4 & C_2 & 2\sigma_\mathrm{v} & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E} & 2 & 0 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

$\,$

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

$\,$

$$\begin{array}{c|cccccc|cc} \hline C_\mathrm{6v} & E & 2C_6 & 2C_3 & C_2 & 3\sigma_\mathrm{v} & 3\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2,z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_1} & 2 & 1 & -1 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \mathrm{E_2} & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \hline \end{array}$$

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$C_{n\mathrm{v}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccc|cc} \hline C_\mathrm{2v} & E & C_2 & \sigma_\mathrm{v}(xz) & \sigma_\mathrm{v}'(yz) & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & z & x^2, y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & x, R_y & xz \\ \mathrm{B_2} & 1 & -1 & -1 & 1 & y, R_x & yz \\ \hline \end{array}$$

$\,$

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

$\,$

$$\begin{array}{c|ccccc|cc} \hline C_\mathrm{4v} & E & 2C_4 & C_2 & 2\sigma_\mathrm{v} & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E} & 2 & 0 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

$\,$

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

$\,$

$$\begin{array}{c|cccccc|cc} \hline C_\mathrm{6v} & E & 2C_6 & 2C_3 & C_2 & 3\sigma_\mathrm{v} & 3\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2,z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_1} & 2 & 1 & -1 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \mathrm{E_2} & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \hline \end{array}$$

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$C_{n\mathrm{v}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccc|cc} \hline C_\mathrm{2v} & E & C_2 & \sigma_\mathrm{v}(xz) & \sigma_\mathrm{v}'(yz) & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & z & x^2, y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & x, R_y & xz \\ \mathrm{B_2} & 1 & -1 & -1 & 1 & y, R_x & yz \\ \hline \end{array}$$

$\,$

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

$\,$

$$\begin{array}{c|ccccc|cc} \hline C_\mathrm{4v} & E & 2C_4 & C_2 & 2\sigma_\mathrm{v} & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E} & 2 & 0 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

$\,$

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

$\,$

$$\begin{array}{c|cccccc|cc} \hline C_\mathrm{6v} & E & 2C_6 & 2C_3 & C_2 & 3\sigma_\mathrm{v} & 3\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2,z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_1} & 2 & 1 & -1 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \mathrm{E_2} & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \hline \end{array}$$

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$C_{n\mathrm{v}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccc|cc} \hline C_\mathrm{2v} & E & C_2 & \sigma_\mathrm{v}(xz) & \sigma_\mathrm{v}'(yz) & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & z & x^2, y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & x, R_y & xz \\ \mathrm{B_2} & 1 & -1 & -1 & 1 & y, R_x & yz \\ \hline \end{array}$$

$\,$

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

$\,$

$$\begin{array}{c|ccccc|cc} \hline C_\mathrm{4v} & E & 2C_4 & C_2 & 2\sigma_\mathrm{v} & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E} & 2 & 0 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

$\,$

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

$\,$

$$\begin{array}{c|cccccc|cc} \hline C_\mathrm{6v} & E & 2C_6 & 2C_3 & C_2 & 3\sigma_\mathrm{v} & 3\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2,z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_1} & 2 & 1 & -1 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \mathrm{E_2} & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \hline \end{array}$$

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$C_{n\mathrm{v}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccc|cc} \hline C_\mathrm{2v} & E & C_2 & \sigma_\mathrm{v}(xz) & \sigma_\mathrm{v}'(yz) & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & z & x^2, y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & x, R_y & xz \\ \mathrm{B_2} & 1 & -1 & -1 & 1 & y, R_x & yz \\ \hline \end{array}$$

$\,$

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

$\,$

$$\begin{array}{c|ccccc|cc} \hline C_\mathrm{4v} & E & 2C_4 & C_2 & 2\sigma_\mathrm{v} & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E} & 2 & 0 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

$\,$

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

$\,$

$$\begin{array}{c|cccccc|cc} \hline C_\mathrm{6v} & E & 2C_6 & 2C_3 & C_2 & 3\sigma_\mathrm{v} & 3\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2,z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_1} & 2 & 1 & -1 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \mathrm{E_2} & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \hline \end{array}$$

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$C_{n\mathrm{v}} (2 \leq n \leq 6)$

$$\begin{array}{c|cccc|cc} \hline C_\mathrm{2v} & E & C_2 & \sigma_\mathrm{v}(xz) & \sigma_\mathrm{v}'(yz) & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & z & x^2, y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & -1 & -1 & R_z & xy \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & x, R_y & xz \\ \mathrm{B_2} & 1 & -1 & -1 & 1 & y, R_x & yz \\ \hline \end{array}$$

$\,$

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

$\,$

$$\begin{array}{c|ccccc|cc} \hline C_\mathrm{4v} & E & 2C_4 & C_2 & 2\sigma_\mathrm{v} & 2\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2, z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & 1 & -1 & & x^2-y^2 \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & 1 & & xy \\ \mathrm{E} & 2 & 0 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \hline \end{array}$$

$\,$

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

$\,$

$$\begin{array}{c|cccccc|cc} \hline C_\mathrm{6v} & E & 2C_6 & 2C_3 & C_2 & 3\sigma_\mathrm{v} & 3\sigma_\mathrm{d} & & \\ \hline \mathrm{A_1} & 1 & 1 & 1 & 1 & 1 & 1 & z & x^2+y^2,z^2 \\ \mathrm{A_2} & 1 & 1 & 1 & 1 & -1 & -1 & R_z & \\ \mathrm{B_1} & 1 & -1 & 1 & -1 & 1 & -1 & & \\ \mathrm{B_2} & 1 & -1 & 1 & -1 & -1 & 1 & & \\ \mathrm{E_1} & 2 & 1 & -1 & -2 & 0 & 0 & (x,y),(R_x,R_y) & (xz,yz) \\ \mathrm{E_2} & 2 & -1 & -1 & 2 & 0 & 0 & & (x^2-y^2,xy) \\ \hline \end{array}$$

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