SHG Tensors under 58 BW MPGs

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$M_0$	$G_0$	$S_0$	$R'$	$G_0$ $\chi_{ij}^{even}$	$G_1$ $\chi_{ij}^{odd}$	LMO
$\overline{1}'$ (2.3.5)	$\overline1$	$1$	$\overline{1}'$	$\overline{1}$ $\chi \text{=}0$	$1$ ${{\left( \begin{matrix} {{\chi}_{11}} & {{\chi}_{12}} & {{\chi}_{13}} & {{\chi}_{14}} & {{\chi}_{15}} & {{\chi}_{16}} \\ {{\chi}_{21}} & {{\chi}_{22}} & {{\chi}_{23}} & {{\chi}_{24}} & {{\chi}_{25}} & {{\chi}_{26}} \\ {{\chi}_{31}} & {{\chi}_{32}} & {{\chi}_{33}} & {{\chi}_{34}} & {{\chi}_{35}} & {{\chi}_{36}} \end{matrix}\right)}_{18}}$	$\times$
$2'$ $y$ ) (3.3.8)	$2$	$1$	$2_{010}'$	$2$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi }_{14}} & 0 & {{\chi }_{16}} \\ {{\chi }_{21}} & {{\chi }_{22}} & {{\chi }_{23}} & 0 & {{\chi }_{25}} & 0 \\ 0 & 0 & 0 & {{\chi }_{34}} & 0 & {{\chi }_{36}} \end{matrix} \right)}_{8}}$	$m$ ${{\left( \begin{matrix} {{\chi }_{11}} & {{\chi }_{12}} & {{\chi }_{13}} & 0 & {{\chi }_{15}} & 0 \\ 0 & 0 & 0 & {{\chi }_{24}} & 0 & {{\chi }_{26}} \\ {{\chi }_{31}} & {{\chi }_{32}} & {{\chi }_{33}} & 0 & {{\chi }_{35}} & 0 \end{matrix} \right)}_{10}}$	$\checkmark$
$m'$ $m$ $\bot$ $y$ ) (4.3.11)	$m$	$1$	$m_{010}'$	$m$ ${{\left( \begin{matrix} {{\chi }_{11}} & {{\chi }_{12}} & {{\chi }_{13}} & 0 & {{\chi }_{15}} & 0 \\ 0 & 0 & 0 & {{\chi }_{24}} & 0 & {{\chi }_{26}} \\ {{\chi }_{31}} & {{\chi }_{32}} & {{\chi }_{33}} & 0 & {{\chi }_{35}} & 0 \end{matrix} \right)}_{10}}$	$2$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi }_{14}} & 0 & {{\chi }_{16}} \\ {{\chi }_{21}} & {{\chi }_{22}} & {{\chi }_{23}} & 0 & {{\chi }_{25}} & 0 \\ 0 & 0 & 0 & {{\chi }_{34}} & 0 & {{\chi }_{36}} \end{matrix} \right)}_{8}}$	$\checkmark$
$2'/m$ $y$ ) (5.3.14)	$2/m$	$m$	$\overline{1}'$	$2/m$ $\chi \text{=}0$	$m$ ${{\left( \begin{matrix} {{\chi }_{11}} & {{\chi }_{12}} & {{\chi }_{13}} & 0 & {{\chi }_{15}} & 0 \\ 0 & 0 & 0 & {{\chi }_{24}} & 0 & {{\chi }_{26}} \\ {{\chi }_{31}} & {{\chi }_{32}} & {{\chi }_{33}} & 0 & {{\chi }_{35}} & 0 \end{matrix} \right)}_{10}}$	$\times$
$2/m'$ $y$ ) (5.4.15)	$2/m$	$2$	$\overline{1}'$	$2/m$ $\chi \text{=}0$	$2$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi }_{14}} & 0 & {{\chi }_{16}} \\ {{\chi }_{21}} & {{\chi }_{22}} & {{\chi }_{23}} & 0 & {{\chi }_{25}} & 0 \\ 0 & 0 & 0 & {{\chi }_{34}} & 0 & {{\chi }_{36}} \end{matrix} \right)}_{8}}$	$\times$
$2'/m'$ $y$ ) (5.5.16)	$2/m$	$\overline{1}$	$2'$	$2/m$ $\chi \text{=}0$	$2/m$ $\chi \text{=}0$	$\checkmark$
$2'2'2$ (6.3.19)	$222$	$2$ $2//z$ )	$2_{100}'$	$222$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi }_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\chi }_{25}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\chi }_{36}} \end{matrix} \right)}_{3}}$	$mm2$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi }_{15}} & 0 \\ 0 & 0 & 0 & {{\chi }_{24}} & 0 & 0 \\ {{\chi }_{31}} & {{\chi }_{32}} & {{\chi }_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{5}}$	$\checkmark$
$m'm2'$ $z$ ) (7.3.22)	$mm2$	$m$ $m{\bot}y$ )	$2_{001}'$	$mm2$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi }_{15}} & 0 \\ 0 & 0 & 0 & {{\chi }_{24}} & 0 & 0 \\ {{\chi }_{31}} & {{\chi }_{32}} & {{\chi }_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{5}}$	$2mm$ ${{\left( \begin{matrix} {{\chi }_{11}} & {{\chi }_{12}} & {{\chi }_{13}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\chi }_{26}} \\ 0 & 0 & 0 & 0 & {{\chi }_{35}} & 0 \end{matrix} \right)}_{5}}$	$\checkmark$
$m'm'2$ $z$ ) (7.4.23)	$mm2$	$2$ $2//z$ )	$m_{100}'$	$mm2$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi }_{15}} & 0 \\ 0 & 0 & 0 & {{\chi }_{24}} & 0 & 0 \\ {{\chi }_{31}} & {{\chi }_{32}} & {{\chi }_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{5}}$	$222$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi }_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\chi }_{25}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\chi }_{36}} \end{matrix} \right)}_{3}}$	$\checkmark$
$mmm'$ (8.3.26)	$mmm$	$mm2$	$\overline{1}'$	$mmm$ $\chi \text{=}0$	$mm2$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi }_{15}} & 0 \\ 0 & 0 & 0 & {{\chi }_{24}} & 0 & 0 \\ {{\chi }_{31}} & {{\chi }_{32}} & {{\chi }_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{5}}$	$\times$
$m'm'm$ (8.4.27)	$mmm$	$2/m$ $2//z$ )	$m_{100}'$	$mmm$ $\chi \text{=}0$	$mmm$ $\chi \text{=}0$	$\checkmark$
$m'm'm'$ (8.5.28)	$mmm$	$222$	$\overline{1}'$	$mmm$ $\chi \text{=}0$	$222$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi }_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\chi }_{25}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\chi }_{36}} \end{matrix} \right)}_{3}}$	$\times$
$4'$ (9.3.31)	$4$	$2$ $2//z$ )	$4_{001}'$	$4$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{15}} & -{{\chi}_{14}} & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{4}}$	$\overline{4}$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & -{{\chi}_{15}} & {{\chi}_{14}} & 0 \\ {{\chi}_{33}} & -{{\chi}_{31}} & 0 & 0 & 0 & {{\chi}_{36}} \end{matrix}\right)}_{4}}$	$\times$
$\overline{4}'$ (10.3.34)	$\overline4$	$2$ $2//z$ )	$\overline{4}_{001}'$	$\overline{4}$ ${{\left( \begin{array}{*{35}{r}} 0 & 0 & 0 & {{\chi}_{14}} & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & -{{\chi}_{15}} & {{\chi}_{14}} & 0 \\ {{\chi}_{33}} & -{{\chi}_{31}} & 0 & 0 & 0 & {{\chi}_{36}} \end{array} \right)}_{4}}$	$4$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{15}} & -{{\chi}_{14}} & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{4}}$	$\times$
$4'/m$ (11.3.37)	$4/m$	$2/m$	$4_{001}'$	$4/m$ $\chi \text{=}0$	$4/m$ $\chi \text{=}0$	$\times$
$4/m'$ (11.4.38)	$4/m$	$4$	$\overline{1}'$	$4/m$ $\chi \text{=}0$	$4$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{15}} & -{{\chi}_{14}} & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{4}}$	$\times$
$4'/m'$ (11.5.39)	$4/m$	$\overline{4}$	$\overline{1}'$	$4/m$ $\chi \text{=}0$	$\overline{4}$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & -{{\chi}_{15}} & {{\chi}_{14}} & 0 \\ {{\chi}_{33}} & -{{\chi}_{31}} & 0 & 0 & 0 & {{\chi}_{36}} \end{matrix}\right)}_{4}}$	$\times$
$4'22'$ (12.3.42)	$422$	$222$	$4_{001}'$	$422$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & -{{\chi}_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{1}}$	$\overline{4}2m$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi }_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\chi }_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\chi }_{36}} \end{matrix} \right)}_{2}}$	$\times$
$42'2'$ (12.4.43)	$422$	$4$	$2_{100}'$	$422$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & -{{\chi}_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{1}}$	$4mm$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{15}} & 0 & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{3}}$	$\checkmark$
$4'm'm$ (13.3.46)	$4mm$	$2mm$	$4_{001}'$	$4mm$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{15}} & 0 & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{3}}$	$\overline{4}2m$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi }_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\chi }_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\chi }_{36}} \end{matrix} \right)}_{2}}$	$\times$
$4m'm'$ (13.4.47)	$4mm$	$4$	$m_{100}'$	$4mm$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{15}} & 0 & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{3}}$	$422$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & -{{\chi}_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{1}}$	$\checkmark$
$\overline{4}'2'm$ (14.3.50)	$\overline42m$	$2mm$	$\overline{4}'$	$\overline{4}2m$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi }_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\chi }_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\chi }_{36}} \end{matrix} \right)}_{2}}$	$4mm$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{15}} & 0 & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{3}}$	$\times$
$\overline{4}'2m'$ (14.4.51)	$\overline42m$	$222$	$\overline{4}'$	$\overline{4}2m$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi }_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\chi }_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\chi }_{36}} \end{matrix} \right)}_{2}}$	$422$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & -{{\chi}_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{1}}$	$\times$
$\overline{4}2'm'$ (14.5.52)	$\overline42m$	$\overline{4}$	$2_{100}'$	$\overline{4}2m$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi }_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\chi }_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\chi }_{36}} \end{matrix} \right)}_{2}}$	$\overline{4}m2$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi }_{15}} & 0 \\ 0 & 0 & 0 & -{{\chi }_{15}} & 0 & 0 \\ {{\chi }_{31}} & -{{\chi }_{31}} & 0 & 0 & 0 & 0 \end{matrix} \right)}_{2}}$	$\checkmark$
$4/m'mm$ (15.3.55)	$4/mmm$	$4mm$	$\overline{1}'$	$4/mmm$ $\chi \text{=}0$	$4mm$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{15}} & 0 & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{3}}$	$\times$
$4'/mm'm$ (15.4.56)	$4/mmm$	$mmm$	$\overline{4}'$	$4/mmm$ $\chi \text{=}0$	$4/mmm$ $\chi \text{=}0$	$\times$
$4'/m'm'm$ (15.5.57)	$4/mmm$	$\overline{4}2m$	$\overline{1}'$	$4/mmm$ $\chi \text{=}0$	$\overline{4}2m$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi }_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\chi }_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\chi }_{36}} \end{matrix} \right)}_{2}}$	$\times$
$4/mm'm'$ (15.6.58)	$4/mmm$	$4/m$	$2_{100}'$	$4/mmm$ $\chi \text{=}0$	$4/mmm$ $\chi \text{=}0$	$\checkmark$
$4/m'm'm'$ (15.7.59)	$4/mmm$	$422$	$\overline{1}'$	$4/mmm$ $\chi \text{=}0$	$422$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & -{{\chi}_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{1}}$	$\times$
$\overline{3}'$ (17.3.64)	$\overline3$	$3$	$\overline{1}'$	$\overline{3}$ $\chi \text{=}0$	$3$ ${{\left( \begin{matrix} {{\chi}_{11}} & -{{\chi}_{11}} & 0 & {{\chi}_{14}} & {{\chi}_{15}} & -{{\chi}_{22}} \\ -{{\chi}_{22}} & {{\chi}_{22}} & 0 & {{\chi}_{15}} & -{{\chi}_{14}} & -{{\chi}_{11}} \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \\\end{matrix} \right)}_{6}}$	$\times$
$32'$ (18.3.67)	$32$	$3$	$2_{100}'$	$32$ ${{\left( \begin{matrix} {{\chi }_{11}} & -{{\chi }_{11}} & 0 & {{\chi }_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & -{{\chi }_{14}} & -{{\chi }_{11}} \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{2}}$	$3m$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi}_{15}} & -{{\chi}_{22}} \\ -{{\chi}_{22}} & {{\chi}_{22}} & 0 & {{\chi}_{15}} & 0 & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{4}}$	$\checkmark$
$3m'$ (19.3.70)	$3m$	$3$	$m_{100}'$	$3m$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi}_{15}} & -{{\chi}_{22}} \\ -{{\chi}_{22}} & {{\chi}_{22}} & 0 & {{\chi}_{15}} & 0 & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{4}}$	$32$ ${{\left( \begin{matrix} {{\chi }_{11}} & -{{\chi }_{11}} & 0 & {{\chi }_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & -{{\chi }_{14}} & -{{\chi }_{11}} \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{2}}$	$\checkmark$
$\overline{3}'m$ (20.3.73)	$\overline3m$	$3m$	$\overline{1}'$	$\overline{3}m$ $\chi \text{=}0$	$3m$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi}_{15}} & -{{\chi}_{22}} \\ -{{\chi}_{22}} & {{\chi}_{22}} & 0 & {{\chi}_{15}} & 0 & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{4}}$	$\times$
$\overline{3}'m'$ (20.4.74)	$\overline3m$	$32$	$\overline{1}'$	$\overline{3}m$ $\chi \text{=}0$	$32$ ${{\left( \begin{matrix} {{\chi }_{11}} & -{{\chi }_{11}} & 0 & {{\chi }_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & -{{\chi }_{14}} & -{{\chi }_{11}} \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{2}}$	$\times$
$\overline{3}m'$ (20.5.75)	$\overline3m$	$\overline{3}$	$2_{100}'$	$\overline{3}m$ $\chi \text{=}0$	$\overline{3}m$ $\chi \text{=}0$	$\checkmark$
$6'$ (21.3.78)	$6$	$3$	$2'$	$6$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{15}} & -{{\chi}_{14}} & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{4}}$	$\overline{6}$ ${{\left( \begin{matrix} {{\chi}_{11}} & -{{\chi}_{11}} & 0 & 0 & 0 & -{{\chi}_{22}} \\ -{{\chi}_{22}} & {{\chi}_{22}} & 0 & 0 & 0 & -{{\chi}_{11}} \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{2}}$	$\times$
$\overline{6}'$ (22.3.81)	$\overline6$	$3$	$m_{001}'$	$\overline{6}$ ${{\left( \begin{matrix} {{\chi}_{11}} & -{{\chi}_{11}} & 0 & 0 & 0 & -{{\chi}_{22}} \\ -{{\chi}_{22}} & {{\chi}_{22}} & 0 & 0 & 0 & -{{\chi}_{11}} \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{2}}$	$6$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{15}} & -{{\chi}_{14}} & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{4}}$	$\times$
$6'/m$ (23.3.84)	$6/m$	$\overline{6}$	$\overline{1}'$	$6/m$ $\chi \text{=}0$	$\overline{6}$ ${{\left( \begin{matrix} {{\chi}_{11}} & -{{\chi}_{11}} & 0 & 0 & 0 & -{{\chi}_{22}} \\ -{{\chi}_{22}} & {{\chi}_{22}} & 0 & 0 & 0 & -{{\chi}_{11}} \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{2}}$	$\times$
$6/m'$ (23.4.85)	$6/m$	$6$	$\overline{1}'$	$6/m$ $\chi \text{=}0$	$6$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{15}} & -{{\chi}_{14}} & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{4}}$	$\times$
$6'/m'$ (23.5.86)	$6/m$	$\overline{3}$	$m_{001}'$	$6/m$ $\chi \text{=}0$	$6/m$ $\chi \text{=}0$	$\times$
$6'22'$ (24.3.89)	$622$	$32$	$2_{001}'$	$622$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & -{{\chi}_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{1}}$	$\overline{6}2m$ ${{\left( \begin{matrix} {{\chi }_{11}} & -{{\chi }_{11}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -{{\chi }_{11}} \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{1}}$	$\times$
$62'2'$ (24.4.90)	$622$	$6$	$2_{100}'$	$622$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & -{{\chi}_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{1}}$	$6mm$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{15}} & 0 & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{3}}$	$\checkmark$
$6'mm'$ (25.3.93)	$6mm$	$3m$	$2_{001}'$	$6mm$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{15}} & 0 & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{3}}$	$\overline{6}m2$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & 0 & -{{\chi }_{22}} \\ -{{\chi }_{22}} & {{\chi }_{22}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{1}}$	$\times$
$6m'm'$ (25.4.94)	$6mm$	$6$	$m_{100}'$	$6mm$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{15}} & 0 & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{3}}$	$622$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & -{{\chi}_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{1}}$	$\checkmark$
$\overline{6}'m'2$ (26.3.97)	$\overline6m2$	$32$	$m_{001}'$	$\overline{6}m2$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & 0 & -{{\chi }_{22}} \\ -{{\chi }_{22}} & {{\chi }_{22}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{1}}$	$622$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & -{{\chi}_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{1}}$	$\times$
$\overline{6}'m2'$ (26.4.98)	$\overline6m2$	$3m$	$m_{001}'$	$\overline{6}m2$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & 0 & -{{\chi }_{22}} \\ -{{\chi }_{22}} & {{\chi }_{22}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{1}}$	$6mm$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{15}} & 0 & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{3}}$	$\times$
$\overline{6}m'2'$ (26.5.99)	$\overline6m2$	$\overline{6}$	$m_{100}'$	$\overline{6}m2$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & 0 & -{{\chi }_{22}} \\ -{{\chi }_{22}} & {{\chi }_{22}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{1}}$	$\overline{6}2m$ ${{\left( \begin{matrix} {{\chi }_{11}} & -{{\chi }_{11}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -{{\chi }_{11}} \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{1}}$	$\checkmark$
$6/m'mm$ (27.3.102)	$6/mmm$	$6mm$	$\overline{1}'$	$6/mmm$ $\chi \text{=}0$	$6mm$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{15}} & 0 & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \end{matrix} \right)}_{3}}$	$\times$
$6'/mmm'$ (27.4.103)	$6/mmm$	$\overline{6}m2$	$\overline{1}'$	$6/mmm$ $\chi \text{=}0$	$\overline{6}m2$ ${{\left( \begin{matrix} 0 & 0 & 0 & 0 & 0 & -{{\chi }_{22}} \\ -{{\chi }_{22}} & {{\chi }_{22}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{1}}$	$\times$
$6'/m'mm'$ (27.5.104)	$6/mmm$	$\overline{3}m$	$6_{001}'$	$6/mmm$ $\chi \text{=}0$	$6/mmm$ $\chi \text{=}0$	$\times$
$6/mm'm'$ (27.6.105)	$6/mmm$	$6/m$	$2_{100}'$	$6/mmm$ $\chi \text{=}0$	$6/mmm$ $\chi \text{=}0$	$\checkmark$
$6/m'm'm'$ (27.7.106)	$6/mmm$	$622$	$\overline{1}'$	$6/mmm$ $\chi \text{=}0$	$622$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & -{{\chi}_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right)}_{1}}$	$\times$
$m'\overline{3}'$ (29.3.111)	$m\overline3$	$23$	$\overline{1}'$	$m\overline{3}$ $\chi \text{=}0$	$23$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\chi}_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\chi}_{14}} \end{matrix} \right)}_{1}}$	$\times$
$4'32'$ (30.3.114)	$432$	$23$	$4_{001}'$	$432$ $\chi \text{=}0$	$\overline{4}3m$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\chi}_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\chi}_{14}} \end{matrix} \right)}_{1}}$	$\times$
$\overline{4}'3m'$ (31.3.117)	$\overline43m$	$23$	$\overline{4}_{001}'$	$\overline{4}3m$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\chi}_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\chi}_{14}} \end{matrix} \right)}_{1}}$	$432$ $\chi \text{=}0$	$\times$
$m'\overline{3}'m$ (32.3.120)	$m\overline3m$	$\overline{4}3m$	$\overline{1}'$	$m\overline{3}m$ $\chi \text{=}0$	$\overline{4}3m$ ${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\chi}_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\chi}_{14}} \end{matrix} \right)}_{1}}$	$\times$
$m\overline{3}m'$ (32.4.121)	$m\overline3m$	$m\overline{3}$	$4_{100}'$	$m\overline{3}m$ $\chi \text{=}0$	$m\overline{3}m$ $\chi \text{=}0$	$\times$
$m'\overline{3}'m'$ (32.5.122)	$m\overline3m$	$432$	$\overline{1}'$	$m\overline{3}m$ $\chi \text{=}0$	$432$ $\chi \text{=}0$	$\times$

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Notes:

(1) MPG: magnetic point group, LMO: linear magneto-optic effect.

$M_{0}=S_{0}+T(G_{0}-S_{0})=S_{0}+{\color{red}{R'}}S_{0}$ $G_0$ $S_0$ $G_0$ $s=g/2$ $s$ $g$ $S_0$ $G_0$ respectively).

(3) The subscript number of SHG tensor means the independent element number.

(4) The symmetry of LMO effect refer the page "LMO table".

Reference: Classification of second harmonic generation effect in magnetically ordered materials | npj Quantum Materials (nature.com)