Second order axial tensors under 32 original and 32 gray MPGs

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MPG $M_0$$M_0$	Axial tensor $[{\beta }_{ij}{]}_{3\times 3}$$[\beta_{ij}]_{3\times3}$		MPG $M_0$$M_0$	Axial tensor $[{\beta }_{ij}{]}_{3\times 3}$$[\beta_{ij}]_{3\times3}$
$1$$1$(1.1.1) ${11}^{\prime }$$11'$(1.2.2)	$\begin{array}{c}{\left(\begin{array}{ccc}{\beta }_{11}& {\beta }_{12}& {\beta }_{13}\\ {\beta }_{21}& {\beta }_{22}& {\beta }_{23}\\ {\beta }_{31}& {\beta }_{32}& {\beta }_{33}\end{array}\right)}_9\end{array}$${{\left( \begin{matrix} {{\beta }_{11}} & {{\beta }_{12}} & {{\beta }_{13}} \\ {{\beta }_{21}} & {{\beta }_{22}} & {{\beta }_{23}} \\ {{\beta }_{31}} & {{\beta }_{32}} & {{\beta }_{33}} \\\end{matrix} \right)}_{9}}$		$3$$3$(16.1.60) ${31}^{\prime }$$31'$(16.2.61)	$\begin{array}{c}{\left(\begin{array}{ccc}{\beta }_{11}& -{\beta }_{21}& 0\\ {\beta }_{21}& {\beta }_{11}& 0\\ 0& 0& {\beta }_{33}\end{array}\right)}_3\end{array}$${{\left( \begin{matrix} {{\beta }_{11}} & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\ \end{matrix} \right)}_{3}}$
$2$$2$(3.1.6) ${21}^{\prime }$$21'$(3.2.7)	$\begin{array}{c}{\left(\begin{array}{ccc}{\beta }_{11}& 0& {\beta }_{13}\\ 0& {\beta }_{22}& 0\\ {\beta }_{31}& 0& {\beta }_{33}\end{array}\right)}_5\end{array}$${{\left( \begin{matrix} {{\beta }_{11}} & 0 & {{\beta }_{13}} \\ 0 & {{\beta }_{22}} & 0 \\ {{\beta }_{31}} & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{5}}$		$32$$32$(18.1.65) ${321}^{\prime }$$321'$(18.2.66)	$\begin{array}{c}{\left(\begin{array}{ccc}{\beta }_{11}& 0& 0\\ 0& {\beta }_{11}& 0\\ 0& 0& {\beta }_{33}\end{array}\right)}_2\end{array}$${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{2}}$
$m$$m$(4.1.9) $m{1}^{\prime }$$m1'$(4.2.10)	$\begin{array}{c}{\left(\begin{array}{ccc}0& {\beta }_{12}& 0\\ {\beta }_{21}& 0& {\beta }_{23}\\ 0& {\beta }_{32}& 0\end{array}\right)}_4\end{array}$${{\left( \begin{matrix} 0 & {{\beta }_{12}} & 0 \\ {{\beta }_{21}} & 0 & {{\beta }_{23}} \\ 0 & {{\beta }_{32}} & 0 \\\end{matrix} \right)}_{4}}$		$3m$$3m$(19.1.68) $3m{1}^{\prime }$$3m1'$(19.2.69)	$\begin{array}{c}{\left(\begin{array}{ccc}0& -{\beta }_{21}& 0\\ {\beta }_{21}& 0& 0\\ 0& 0& 0\end{array}\right)}_1\end{array}$${{\left( \begin{matrix} 0 & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$
$222$$222$(6.1.17) ${2221}^{\prime }$$2221'$(6.2.18)	$\begin{array}{c}{\left(\begin{array}{ccc}{\beta }_{11}& 0& 0\\ 0& {\beta }_{22}& 0\\ 0& 0& {\beta }_{33}\end{array}\right)}_3\end{array}$${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{22}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{3}}$		$6$$6$(21.1.76) ${61}^{\prime }$$61'$(21.2.77)	$\begin{array}{c}{\left(\begin{array}{ccc}{\beta }_{11}& -{\beta }_{21}& 0\\ {\beta }_{21}& {\beta }_{11}& 0\\ 0& 0& {\beta }_{33}\end{array}\right)}_3\end{array}$${{\left( \begin{matrix} {{\beta }_{11}} & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{3}}$
$mm2$$mm2$(7.1.20) $mm{21}^{\prime }$$mm21'$(7.2.21)	$\begin{array}{c}{\left(\begin{array}{ccc}0& {\beta }_{12}& 0\\ {\beta }_{21}& 0& 0\\ 0& 0& 0\end{array}\right)}_2\end{array}$${{\left( \begin{matrix} 0 & {\beta_{12}} & 0 \\ {\beta_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{2}}$		$\bar{6}$$\overline{6}$(22.1.79) $\bar{6}{1}^{\prime }$$\overline{6}1'$(22.2.80)	$\beta \text{=}0$$\beta \text{=}0$
$4$$4$(9.1.29) ${41}^{\prime }$$41'$(9.2.30)	$\begin{array}{c}{\left(\begin{array}{ccc}{\beta }_{11}& -{\beta }_{21}& 0\\ {\beta }_{21}& {\beta }_{11}& 0\\ 0& 0& {\beta }_{33}\end{array}\right)}_3\end{array}$${{\left( \begin{matrix} {{\beta }_{11}} & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{3}}$		$622$$622$(24.1.87) ${6221}^{\prime }$$6221'$(24.2.88)	$\begin{array}{c}{\left(\begin{array}{ccc}{\beta }_{11}& 0& 0\\ 0& {\beta }_{11}& 0\\ 0& 0& {\beta }_{33}\end{array}\right)}_2\end{array}$${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{2}}$
$\bar{4}$$\overline{4}$(10.1.32) $\bar{4}{1}^{\prime }$$\overline{4}1'$(10.2.33)	$\begin{array}{c}{\left(\begin{array}{ccc}{\beta }_{11}& {\beta }_{21}& 0\\ {\beta }_{21}& -{\beta }_{11}& 0\\ 0& 0& 0\end{array}\right)}_2\end{array}$${{\left( \begin{matrix} {{\beta }_{11}} & {{\beta }_{21}} & 0 \\ {{\beta }_{21}} & -{{\beta }_{11}} & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{2}}$		$6mm$$6mm$(25.1.91) $6mm{1}^{\prime }$$6mm1'$(25.2.92)	$\begin{array}{c}{\left(\begin{array}{ccc}0& -{\beta }_{21}& 0\\ {\beta }_{21}& 0& 0\\ 0& 0& 0\end{array}\right)}_1\end{array}$${{\left( \begin{matrix} 0 & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$
$422$$422$(12.1.40) ${4221}^{\prime }$$4221'$(12.2.41)	$\begin{array}{c}{\left(\begin{array}{ccc}{\beta }_{11}& 0& 0\\ 0& {\beta }_{11}& 0\\ 0& 0& {\beta }_{33}\end{array}\right)}_3\end{array}$${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{33}} \\\end{matrix} \right)}_{3}}$		$\bar{6}m2$$\overline{6}m2$(26.1.95) $\bar{6}m{21}^{\prime }$$\overline{6}m21'$(26.2.96)	$\beta \text{=}0$$\beta \text{=}0$
$4mm$$4mm$(13.1.44) $4mm{1}^{\prime }$$4mm1'$(13.2.45)	$\begin{array}{c}{\left(\begin{array}{ccc}0& -{\beta }_{21}& 0\\ {\beta }_{21}& 0& 0\\ 0& 0& 0\end{array}\right)}_1\end{array}$${{\left( \begin{matrix} 0 & -{{\beta }_{21}} & 0 \\ {{\beta }_{21}} & 0 & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$		$23$$23$(28.1.107) ${231}^{\prime }$$231'$(28.2.108) $432$$432$(30.1.112) ${4321}^{\prime }$$4321'$(30.2.113)	$\begin{array}{c}{\left(\begin{array}{ccc}{\beta }_{11}& 0& 0\\ 0& {\beta }_{11}& 0\\ 0& 0& {\beta }_{11}\end{array}\right)}_1\end{array}$${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & {{\beta }_{11}} & 0 \\ 0 & 0 & {{\beta }_{11}} \\\end{matrix} \right)}_{1}}$
$\bar{4}2m$$\overline{4}2m$(14.1.48) $\bar{4}2m{1}^{\prime }$$\overline{4}2m1'$(14.2.49)	$\begin{array}{c}{\left(\begin{array}{ccc}{\beta }_{11}& 0& 0\\ 0& -{\beta }_{11}& 0\\ 0& 0& 0\end{array}\right)}_1\end{array}$${{\left( \begin{matrix} {{\beta }_{11}} & 0 & 0 \\ 0 & -{{\beta }_{11}} & 0 \\ 0 & 0 & 0 \\\end{matrix} \right)}_{1}}$		$\bar{4}3m$$\overline{4}3m$(31.1.115) $\bar{4}3m{1}^{\prime }$$\overline{4}3m1'$(31.2.116)	$\beta \text{=}0$$\beta \text{=}0$

Notes:

1. Axial tensor is even under the inversion operation. This kind of tensor is suitable for circular photogalvanic effect (CPGE), Edelstein effect, magnetoelectric effect

2. There is only the even part (${\beta }^{\text{even}}$$\beta^{\text{even}}$) for grey MPGs (the second ones of each row, with $1^{\prime }$$1'$), and the odd and even parts (${\beta }^{\text{even}}$$\beta^{\text{even}}$) and ${\beta }^{\text{odd}}$$\beta^{\text{odd}}$) have same symmetry in original MPGs (the first MPGs of each row, without $1^{\prime }$$1'$).

3. The subscript number of the tensor means the independent element number.

4. $[{\beta }_{ij}{]}_{3\times 3}$$[\beta_{ij}]_{3\times3}$ is zero for the original and grey groups not listed here because they have the inversion symmetry.

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