Second order axial tensors under 58 BW MPGs

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$M_0$	$G_0$	$S_0$	$R'$	$G_0$ $[\beta_{ij}^{\text{even}}]_{3\times3}$	$G_1$ $[\beta _{ij}^{\text{odd}}]_{3\times3}$
$\overline{1}’$ (2.3.5)	$\overline1$	$1$	$\overline{1}'$	$\overline{1}$ $\beta \text{=}0$	$1$ ${{\left( \begin{matrix} {{\beta }_{11}} & {{\beta }_{12}} & {{\beta }_{13}} \\ {{\beta }_{21}} & {{\beta }_{22}} & {{\beta }_{23}} \\ {{\beta }_{31}} & {{\beta }_{32}} & {{\beta }_{33}} \\\end{matrix} \right)}_{9}}$
$2’$ $y$ ) (3.3.8)	$2$	$1$	$2_{010}'$	$2$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & {{\beta }_{13}} \\ 0 & {{\beta }_{22}} & 0 \\ {{\beta }_{31}} & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{5}}$	$m$ ${{\left( \begin{matrix} 0 & {{\beta }_{12}} & 0 \\ {{\beta }_{21}} & 0 & {{\beta }_{23}} \\ 0 & {{\beta }_{32}} & 0 \\\end{matrix} \right)}_{4}}$
$m’$ $m$ $\bot$ $y$ ) (4.3.11)	$m$ $m \bot y$ )	$1$	$m_{010}'$	$m$ ${{\left( \begin{matrix} 0 & {{\beta }_{12}} & 0 \\ {{\beta }_{21}} & 0 & {{\beta }_{23}} \\ 0 & {{\beta }_{32}} & 0 \\\end{matrix} \right)}_{4}}$	$2$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & {{\beta }_{13}} \\ 0 & {{\beta }_{22}} & 0 \\ {{\beta }_{31}} & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{5}}$
$2’/m$ $y$ ) (5.3.14)	$2/m$	$m$	$\overline{1}'$	$2/m$ $\beta \text{=}0$	$m$ ${{\left( \begin{matrix} 0 & {{\beta }_{12}} & 0 \\ {{\beta }_{21}} & 0 & {{\beta }_{23}} \\ 0 & {{\beta }_{32}} & 0 \\\end{matrix} \right)}_{4}}$
$2/m’$ $y$ ) (5.4.15)	$2/m$	$2$	$\overline{1}'$	$2/m$ $\beta \text{=}0$	$2$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & {{\beta }_{13}} \\ 0 & {{\beta }_{22}} & 0 \\ {{\beta }_{31}} & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{5}}$
$2’/m’$ $y$ ) (5.5.16)	$2/m$	$\overline{1}$	$2'$	$2/m$ $\beta \text{=}0$	$2/m$ $\beta \text{=}0$
$2’2’2$ (6.3.19)	$222$	$2$ $z$ )	$2_{100}'$	$222$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{22}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{3}}$	$mm2$ ${{\left( \begin{matrix} 0 & {\beta_{12}} & 0 \\ {\beta_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{2}}$
$m’m2’$ (7.3.22)	$mm2$	$m$ $m{\bot}y$ )	$2_{001}'$	$mm2$ ${{\left( \begin{matrix} 0 & {\beta_{12}} & 0 \\ {\beta_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{2}}$	$2mm$ ${{\left( \begin{matrix} 0 & 0 & 0 \\ 0 & 0 & {{\beta }_{32}} \\ 0 & {{\beta }_{23}} & 0 \\\end{matrix} \right)}_{2}}$
$m’m’2$ (7.4.23)	$mm2$	$2$ $z$ )	$m_{100}'$	$mm2$ ${{\left( \begin{matrix} 0 & {\beta_{12}} & 0 \\ {\beta_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{2}}$	$222$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{22}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{3}}$
$mmm’$ (8.3.26)	$mmm$	$mm2$	$\overline{1}'$	$mmm$ $\beta \text{=}0$	$mm2$ ${{\left( \begin{matrix} 0 & {\beta_{12}} & 0 \\ {\beta_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{2}}$
$m’m’m$ (8.4.27)	$mmm$	$2/m$ $2//z$ )	$m_{100}'$	$mmm$ $\beta \text{=}0$	$mmm$ $\beta \text{=}0$
$m’m’m’$ (8.5.28)	$mmm$	$222$	$\overline{1}'$	$mmm$ $\beta \text{=}0$	$222$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{22}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{3}}$
$4’$ (9.3.31)	$4$	$2$ $2//z$ )	$4_{001}'$	$4$ ${{\left( \begin{matrix} {{\beta }_{11}} & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{3}}$	$\overline{4}$ ${{\left( \begin{matrix} {{\beta }_{11}} & {{\beta }_{21}} & 0 \\ {{\beta }_{21}} & -{{\beta }_{11}} & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{2}}$
$\overline{4}’$ (10.3.34)	$\overline4$	$2$ $2//z$ )	$\overline{4}_{001}'$	$\overline{4}$ ${{\left( \begin{matrix} {{\beta }_{11}} & {{\beta }_{21}} & 0 \\ {{\beta }_{21}} & -{{\beta }_{11}} & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{2}}$	$4$ ${{\left( \begin{matrix} {{\beta }_{11}} & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{3}}$
$4'/m$ (11.3.37)	$4/m$	$2/m$	$4_{001}'$	$4/m$ $\beta \text{=}0$	$4/m$ $\beta \text{=}0$
$4/m'$ (11.4.38)	$4/m$	$4$	$\overline{1}'$	$4/m$ $\beta \text{=}0$	$4$ ${{\left( \begin{matrix} {{\beta }_{11}} & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{3}}$
$4'/m'$ (11.5.39)	$4/m$	$\overline{4}$	$\overline{1}'$	$4/m$ $\beta \text{=}0$	$\overline{4}$ ${{\left( \begin{matrix} {{\beta }_{11}} & {{\beta }_{21}} & 0 \\ {{\beta }_{21}} & -{{\beta }_{11}} & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{2}}$
$4'22'$ (12.3.42)	$422$	$222$	$4_{001}'$	$422$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{3}}$	$\overline{4}2m$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & -{{\beta }_{11}} & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$
$42'2'$ (12.4.43)	$422$	$4$	$2_{100}'$	$422$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{3}}$	$4mm$ ${{\left( \begin{matrix} 0 & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$
$4'm'm$ (13.3.46)	$4mm$	$2mm$	$4_{001}'$	$4mm$ ${{\left( \begin{matrix} 0 & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$	$\overline{4}2m$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & -{{\beta }_{11}} & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$
$4m'm'$ (13.4.47)	$4mm$	$4$	$m_{100}'$	$4mm$ ${{\left( \begin{matrix} 0 & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$	$422$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{3}}$
$\overline{4}'2'm$ (14.3.50)	$\overline42m$	$2mm$	$\overline{4}'$	$\overline{4}2m$ ${{\left( \begin{matrix}{{\beta }_{11}} & 0 & 0 \\ 0 & -{{\beta }_{11}} & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$	$4mm$ ${{\left( \begin{matrix} 0 & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$
$\overline{4}'2m'$ (14.4.51)	$\overline42m$	$222$	$\overline{4}'$	$\overline{4}2m$ ${{\left( \begin{matrix}{{\beta }_{11}} & 0 & 0 \\ 0 & -{{\beta }_{11}} & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$	$422$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{3}}$
$\overline{4}2'm'$ (14.5.52)	$\overline42m$	$\overline{4}$	$2_{100}'$	$\overline{4}2m$ ${{\left( \begin{matrix}{{\beta }_{11}} & 0 & 0 \\ 0 & -{{\beta }_{11}} & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$	$\overline{4}m2$ ${{\left( \begin{matrix} 0 & {{\beta }_{12}} & 0 \\ {{\beta }_{12}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$
$4/m'mm$ (15.3.55)	$4/mmm$	$4mm$	$\overline{1}'$	$4/mmm$ $\beta \text{=}0$	$4mm$ ${{\left( \begin{matrix} 0 & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$
$4'/mm'm$ (15.4.56)	$4/mmm$	$mmm$	$\overline{4}'$	$4/mmm$ $\beta \text{=}0$	$4/mmm$ $\beta \text{=}0$
$4'/m'm'm$ (15.5.57)	$4/mmm$	$\overline{4}2m$	$\overline{1}'$	$4/mmm$ $\beta \text{=}0$	$\overline{4}2m$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & -{{\beta }_{11}} & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$
$4/mm'm'$ (15.6.58)	$4/mmm$	$4/m$	$2_{100}'$	$4/mmm$ $\beta \text{=}0$	$4/mmm$ $\beta \text{=}0$
$4/m'm'm'$ (15.7.59)	$4/mmm$	$422$	$\overline{1}'$	$4/mmm$ $\beta \text{=}0$	$422$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{3}}$
$\overline{3}’$ (17.3.64)	$\overline3$	$3$	$\overline{1}'$	$\overline{3}$ $\beta \text{=}0$	$3$ ${{\left( \begin{matrix} {{\beta }_{11}} & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{3}}$
$32'$ (18.3.67)	$32$	$3$	$2_{100}'$	$32$ ${{\left( \begin{matrix}{{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{2}}$	$3m$ ${{\left( \begin{matrix} 0 & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$
$3m’$ (19.3.70)	$3m$	$3$	$m_{100}'$	$3m$ ${{\left( \begin{matrix}0 & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$	$32$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{2}}$
$\overline{3}'m$ (20.3.73)	$\overline3m$	$3m$	$\overline{1}'$	$\overline{3}m$ $\beta \text{=}0$	$3m$ ${{\left( \begin{matrix} 0 & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$
$\overline{3}’m’$ (20.4.74)	$\overline3m$	$32$	$\overline{1}'$	$\overline{3}m$ $\beta \text{=}0$	$32$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{2}}$
$\overline{3}m’$ (20.5.75)	$\overline3m$	$\overline{3}$	$2_{100}'$	$\overline{3}m$ $\beta \text{=}0$	$\overline{3}m$ $\beta \text{=}0$
$6’$ (21.3.78)	$6$	$3$	$2'$	$6$ ${{\left( \begin{matrix}{{\beta }_{11}} & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{3}}$	$\overline{6}$ $\beta \text{=}0$
$\overline{6}'$ (22.3.81)	$\overline6$	$3$	$m_{001}'$	$\overline{6}$ $\beta \text{=}0$	$6$ ${{\left( \begin{matrix} {{\beta }_{11}} & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{3}}$
$6'/m$ (23.3.84)	$6/m$	$\overline{6}$	$\overline{1}'$	$6/m$ $\beta \text{=}0$	$\overline{6}$ $\beta \text{=}0$
$6/m'$ (23.4.85)	$6/m$	$6$	$\overline{1}'$	$6/m$ $\beta \text{=}0$	$6$ ${{\left( \begin{matrix} {{\beta }_{11}} & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{3}}$
$6'/m'$ (23.5.86)	$6/m$	$\overline{3}$	$m_{001}'$	$6/m$ $\beta \text{=}0$	$6/m$ $\beta \text{=}0$
$6'22'$ (24.3.89)	$622$	$32$	$2_{001}'$	$622$ ${{\left( \begin{matrix}{{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{2}}$	$\overline{6}2m$ $\beta \text{=}0$
$62'2'$ (24.4.90)	$622$	$6$	$2_{100}'$	$622$ ${{\left( \begin{matrix}{{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{2}}$	$6mm$ ${{\left( \begin{matrix} 0 & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$
$6'mm'$ (25.3.93)	$6mm$	$3m$	$2_{001}'$	$6mm$ ${{\left( \begin{matrix}0 & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$	$\overline{6}m2$ $\beta \text{=}0$
$6m'm'$ (25.4.94)	$6mm$	$6$	$m_{100}'$	$6mm$ ${{\left( \begin{matrix}0 & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$	$622$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{2}}$
$\overline{6}’m’2$ (26.3.97)	$\overline6m2$	$32$	$m_{001}'$	$\overline{6}m2$ $\beta \text{=}0$	$622$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{2}}$
$\overline{6}’m2’$ (26.4.98)	$\overline6m2$	$3m$	$m_{001}'$	$\overline{6}m2$ $\beta \text{=}0$	$6mm$ ${{\left( \begin{matrix} 0 & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$
$\overline{6}m’2’$ (26.5.99)	$\overline6m2$	$\overline{6}$	$m_{100}'$	$\overline{6}m2$ $\beta \text{=}0$	$\overline{6}2m$ $\beta \text{=}0$
$6/m'mm$ (27.3.102)	$6/mmm$	$6mm$	$\overline{1}'$	$6/mmm$ $\beta \text{=}0$	$6mm$ ${{\left( \begin{matrix} 0 & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$
$6'/mmm'$ (27.4.103)	$6/mmm$	$\overline{6}m2$	$\overline{1}'$	$6/mmm$ $\beta \text{=}0$	$\overline{6}m2$ $\beta \text{=}0$
$6'/m'mm'$ (27.5.104)	$6/mmm$	$\overline{3}m$	$6_{001}'$	$6/mmm$ $\beta \text{=}0$	$6/mmm$ $\beta \text{=}0$
$6/mm'm'$ (27.6.105)	$6/mmm$	$6/m$	$2_{100}'$	$6/mmm$ $\beta \text{=}0$	$6/mmm$ $\beta \text{=}0$
$6/m'm'm'$ (27.7.106)	$6/mmm$	$622$	$\overline{1}'$	$6/mmm$ $\beta \text{=}0$	$622$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{2}}$
$m'\overline{3}'$ (29.3.111)	$m\overline3$	$23$	$\overline{1}'$	$m\overline{3}$ $\beta \text{=}0$	$23$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{11}} \\\end{matrix} \right)}_{1}}$
$4'32'$ (30.3.114)	$432$	$23$	$4_{001}'$	$432$ ${{\left( \begin{matrix}{{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{11}} \\\end{matrix} \right)}_{1}}$	$\overline{4}3m$ $\beta \text{=}0$
$\overline{4}'3m'$ (31.3.117)	$\overline43m$	$23$	$\overline{4}_{001}'$	$\overline{4}3m$ $\beta \text{=}0$	$432$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{11}} \\\end{matrix} \right)}_{1}}$
$m'\overline{3}'m$ (32.3.120)	$m\overline3m$	$\overline{4}3m$	$\overline{1}'$	$m\overline{3}m$ $\beta \text{=}0$	$\overline{4}3m$ $\beta \text{=}0$
$m\overline{3}m'$ (32.4.121)	$m\overline3m$	$m\overline{3}$	$4_{100}'$	$m\overline{3}m$ $\beta \text{=}0$	$m\overline{3}m$ $\beta \text{=}0$
$m'\overline{3}'m'$ (32.5.122)	$m\overline3m$	$432$	$\overline{1}'$	$m\overline{3}m$ $\beta \text{=}0$	$432$ ${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{11}} \\\end{matrix} \right)}_{1}}$

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Notes:

Axial tensor is even under the inversion operation. This kind of tensor is suitable for circular photogalvanic effect (CPGE), Edelstein effect, magnetoelectric effect.
The subscript number of the tensor means the independent element number.

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