SHG tensors under 32 original and 32 gray MPGs

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MPG	SHG tensor	LMO
$1$ (1.1.1) $11'$ (1.2.2)	${{\left( \begin{matrix} {{\chi}_{11}} & {{\chi}_{12}} & {{\chi}_{13}} & {{\chi}_{14}} & {{\chi}_{15}} & {{\chi}_{16}} \\ {{\chi}_{21}} & {{\chi}_{22}} & {{\chi}_{23}} & {{\chi}_{24}} & {{\chi}_{25}} & {{\chi}_{26}} \\ {{\chi}_{31}} & {{\chi}_{32}} & {{\chi}_{33}} & {{\chi}_{34}} & {{\chi}_{35}} & {{\chi}_{36}} \\\end{matrix} \right)}_{18}}$	$\checkmark$
$\overline1$ (2.1.3) $\overline11'$ (2.2.4)	$\chi \text{=}0$	$\checkmark$
$2$ (3.1.6) $21'$ (3.2.7)	${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi }_{14}} & 0 & {{\chi }_{16}} \\ {{\chi }_{21}} & {{\chi }_{22}} & {{\chi }_{23}} & 0 & {{\chi }_{25}} & 0 \\ 0 & 0 & 0 & {{\chi }_{34}} & 0 & {{\chi }_{36}} \\\end{matrix} \right)}_{8}}$	$\checkmark$
$m$ (4.1.9) $m1'$ (4.2.10)	${{\left( \begin{matrix} {{\chi }_{11}} & {{\chi }_{12}} & {{\chi }_{13}} & 0 & {{\chi }_{15}} & 0 \\ 0 & 0 & 0 & {{\chi }_{24}} & 0 & {{\chi }_{26}} \\ {{\chi }_{31}} & {{\chi }_{32}} & {{\chi }_{33}} & 0 & {{\chi }_{35}} & 0 \\\end{matrix} \right)}_{10}}$	$\checkmark$
$2/m$ (5.1.12) $2/m1'$ (5.2.13)	$\chi \text{=}0$	$\checkmark$
$222$ (6.1.17) $2221'$ (6.2.18)	${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi }_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\chi }_{25}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\chi }_{36}} \\\end{matrix} \right)}_{3}}$	$\times$
$mm2$ (7.1.20) $mm21'$ (7.2.21)	${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{24}} & 0 & 0 \\ {{\chi}_{31}} & {{\chi}_{32}} & {{\chi}_{33}} & 0 & 0 & 0 \\\end{matrix} \right)}_{5}}$	$\times$
$mmm$ (8.1.24) $mmm1'$ (8.2.25)	$\chi \text{=}0$	$\times$
$4$ (9.1.29) $41'$ (9.2.30)	${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{15}} & -{{\chi}_{14}} & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \\\end{matrix} \right)}_{4}}$	$\checkmark$
$\overline4$ (10.1.32) $\overline41'$ (10.2.33)	${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & -{{\chi}_{15}} & {{\chi}_{14}} & 0 \\ {{\chi}_{33}} & -{{\chi}_{31}} & 0 & 0 & 0 & {{\chi}_{36}} \\\end{matrix} \right)}_{4}}$	$\checkmark$
$4/m$ (11.1.35) $4/m1'$ (11.2.36)	$\chi \text{=}0$	$\checkmark$
$422$ (12.1.40) $4221'$ (12.2.41)	${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & -{{\chi}_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$	$\times$
$4mm$ (13.1.44) $4mm1'$ (13.2.45)	${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{15}} & 0 & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \\\end{matrix} \right)}_{3}}$	$\times$
$\overline42m$ (14.1.48) $\overline42m1'$ (14.2.49)	${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi }_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\chi }_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\chi }_{36}} \\ \end{matrix} \right)}_{2}}$	$\times$
$4/mmm$ (15.1.53) $4/mmm1'$ (15.2.54)	$\chi \text{=}0$	$\times$
$3$ (16.1.60) $31'$ (16.2.61)	${{\left( \begin{matrix} {{\chi}_{11}} & -{{\chi}_{11}} & 0 & {{\chi}_{14}} & {{\chi}_{15}} & -{{\chi}_{22}} \\ -{{\chi}_{22}} & {{\chi}_{22}} & 0 & {{\chi}_{15}} & -{{\chi}_{14}} & -{{\chi}_{11}} \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \\\end{matrix} \right)}_{6}}$	$\checkmark$
$\overline3$ (17.1.62) $\overline31'$ (17.2.63)	$\chi \text{=}0$	$\checkmark$
$32$ (18.1.65) $321'$ (18.2.66)	${{\left( \begin{matrix} {{\chi }_{11}} & -{{\chi }_{11}} & 0 & {{\chi }_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & -{{\chi }_{14}} & -{{\chi }_{11}} \\ 0 & 0 & 0 & 0 & 0 & 0 \\\end{matrix} \right)}_{2}}$	$\times$
$3m$ (19.1.68) $3m1'$ (19.2.69)	${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi}_{15}} & -{{\chi}_{22}} \\ -{{\chi}_{22}} & {{\chi}_{22}} & 0 & {{\chi}_{15}} & 0 & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \\\end{matrix} \right)}_{4}}$	$\times$
$\overline3m$ (20.1.71) $\overline3m1'$ (20.2.72)	$\chi \text{=}0$	$\times$
$6$ (21.1.76) $61'$ (21.2.77)	${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{15}} & -{{\chi}_{14}} & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \\\end{matrix} \right)}_{4}}$	$\checkmark$
$\overline6$ (22.1.79) $\overline61'$ (22.2.80)	${{\left( \begin{matrix} {{\chi}_{11}} & -{{\chi}_{11}} & 0 & 0 & 0 & -{{\chi}_{22}} \\ -{{\chi}_{22}} & {{\chi}_{22}} & 0 & 0 & 0 & -{{\chi}_{11}} \\ 0 & 0 & 0 & 0 & 0 & 0 \\\end{matrix} \right)}_{2}}$	$\checkmark$
$6/m$ (23.1.82) $6/m1'$ (23.2.83)	$\chi \text{=}0$	$\checkmark$
$622$ (24.1.87) $6221'$ (24.2.88)	${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & -{{\chi}_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$	$\times$
$6mm$ (25.1.91) $6mm1'$ (25.2.92)	${{\left( \begin{matrix} 0 & 0 & 0 & 0 & {{\chi}_{15}} & 0 \\ 0 & 0 & 0 & {{\chi}_{15}} & 0 & 0 \\ {{\chi}_{31}} & {{\chi}_{31}} & {{\chi}_{33}} & 0 & 0 & 0 \\\end{matrix} \right)}_{3}}$	$\times$
$\overline6m2$ (26.1.95) $\overline6m21'$ (26.2.96)	${{\left( \begin{matrix} 0 & 0 & 0 & 0 & 0 & -{{\chi}_{22}} \\ -{{\chi}_{22}} & {{\chi}_{22}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$	$\times$
$6/mmm$ (27.1.100) $6/mmm1'$ (27.2.101)	$\chi \text{=}0$	$\times$
$23$ (28.1.107) $231'$ (28.2.108)	${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\chi}_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\chi}_{14}} \\\end{matrix} \right)}_{1}}$	$\times$
$m\overline3$ (29.1.109) $m\overline31'$ (29.2.110)	$\chi \text{=}0$	$\times$
$432$ (30.1.112) $4321'$ (30.2.113)	$\chi \text{=}0$	$\times$
$\overline43m$ (31.1.115) $\overline43m1'$ (31.2.116)	${{\left( \begin{matrix} 0 & 0 & 0 & {{\chi}_{14}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {{\chi}_{14}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {{\chi}_{14}} \\\end{matrix} \right)}_{1}}$	$\times$
$m\overline3m$ (32.1.118) $m\overline3m1'$ (32.2.119)	$\chi \text{=}0$	$\times$

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Notes:

$\chi^{even}$ $1'$ $\chi^{even}$ $\chi^{odd}$ $1'$ ). (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)