| |$\rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})) \subset \operatorname{Sym}_{5}$| | |$\Gamma \simeq \operatorname{Gal}(L/\textbf{k})$| | |$G=\operatorname{Aut}_{\textbf{k}}(X)$| | |
| 2e | |$\langle (12)(34) \rangle \times \langle (13)(24) \rangle $| | |$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$| | |$\langle \widehat{\alpha } \rangle \times \langle \widehat{\beta } \rangle \simeq \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$|, where |$\widehat{\alpha }$| and |$\widehat{\beta }$| are the lifts of involutions of |$\mathbb{P}^{2}$| |
| 2g | |$\langle (1234) \rangle $| | |$\mathbb{Z}/4\mathbb{Z}$| | |$\langle \widehat{\alpha } \rangle \simeq \mathbb{Z}/4\mathbb{Z}$|, where |$\widehat{\alpha }$| is the lift of an automorphism of |$\mathbb{P}^{2}$| of order four |
| 2i | |$\langle (1234),(13) \rangle $| | |$\operatorname{D}_{4}$| | |$\langle \widehat{\alpha } \rangle \simeq \mathbb{Z}/2\mathbb{Z}$|, where |$\widehat{\alpha }$| is the lift of an involution of |$\mathbb{P}^{2}$| |
| 2h | |$\langle (12)(34),(123) \rangle $| | |$\mathcal{A}_{4}$| | |$\lbrace \operatorname{id} \rbrace $| |
| 2j | |$\langle (1234),(12) \rangle $| | |$\operatorname{Sym}_{4}$| | |$\lbrace \operatorname{id} \rbrace $| |
| |$\rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})) \subset \operatorname{Sym}_{5}$| | |$\Gamma \simeq \operatorname{Gal}(L/\textbf{k})$| | |$G=\operatorname{Aut}_{\textbf{k}}(X)$| | |
| 2e | |$\langle (12)(34) \rangle \times \langle (13)(24) \rangle $| | |$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$| | |$\langle \widehat{\alpha } \rangle \times \langle \widehat{\beta } \rangle \simeq \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$|, where |$\widehat{\alpha }$| and |$\widehat{\beta }$| are the lifts of involutions of |$\mathbb{P}^{2}$| |
| 2g | |$\langle (1234) \rangle $| | |$\mathbb{Z}/4\mathbb{Z}$| | |$\langle \widehat{\alpha } \rangle \simeq \mathbb{Z}/4\mathbb{Z}$|, where |$\widehat{\alpha }$| is the lift of an automorphism of |$\mathbb{P}^{2}$| of order four |
| 2i | |$\langle (1234),(13) \rangle $| | |$\operatorname{D}_{4}$| | |$\langle \widehat{\alpha } \rangle \simeq \mathbb{Z}/2\mathbb{Z}$|, where |$\widehat{\alpha }$| is the lift of an involution of |$\mathbb{P}^{2}$| |
| 2h | |$\langle (12)(34),(123) \rangle $| | |$\mathcal{A}_{4}$| | |$\lbrace \operatorname{id} \rbrace $| |
| 2j | |$\langle (1234),(12) \rangle $| | |$\operatorname{Sym}_{4}$| | |$\lbrace \operatorname{id} \rbrace $| |
| |$\rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})) \subset \operatorname{Sym}_{5}$| | |$\Gamma \simeq \operatorname{Gal}(L/\textbf{k})$| | |$G=\operatorname{Aut}_{\textbf{k}}(X)$| | |
| 2e | |$\langle (12)(34) \rangle \times \langle (13)(24) \rangle $| | |$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$| | |$\langle \widehat{\alpha } \rangle \times \langle \widehat{\beta } \rangle \simeq \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$|, where |$\widehat{\alpha }$| and |$\widehat{\beta }$| are the lifts of involutions of |$\mathbb{P}^{2}$| |
| 2g | |$\langle (1234) \rangle $| | |$\mathbb{Z}/4\mathbb{Z}$| | |$\langle \widehat{\alpha } \rangle \simeq \mathbb{Z}/4\mathbb{Z}$|, where |$\widehat{\alpha }$| is the lift of an automorphism of |$\mathbb{P}^{2}$| of order four |
| 2i | |$\langle (1234),(13) \rangle $| | |$\operatorname{D}_{4}$| | |$\langle \widehat{\alpha } \rangle \simeq \mathbb{Z}/2\mathbb{Z}$|, where |$\widehat{\alpha }$| is the lift of an involution of |$\mathbb{P}^{2}$| |
| 2h | |$\langle (12)(34),(123) \rangle $| | |$\mathcal{A}_{4}$| | |$\lbrace \operatorname{id} \rbrace $| |
| 2j | |$\langle (1234),(12) \rangle $| | |$\operatorname{Sym}_{4}$| | |$\lbrace \operatorname{id} \rbrace $| |
| |$\rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})) \subset \operatorname{Sym}_{5}$| | |$\Gamma \simeq \operatorname{Gal}(L/\textbf{k})$| | |$G=\operatorname{Aut}_{\textbf{k}}(X)$| | |
| 2e | |$\langle (12)(34) \rangle \times \langle (13)(24) \rangle $| | |$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$| | |$\langle \widehat{\alpha } \rangle \times \langle \widehat{\beta } \rangle \simeq \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$|, where |$\widehat{\alpha }$| and |$\widehat{\beta }$| are the lifts of involutions of |$\mathbb{P}^{2}$| |
| 2g | |$\langle (1234) \rangle $| | |$\mathbb{Z}/4\mathbb{Z}$| | |$\langle \widehat{\alpha } \rangle \simeq \mathbb{Z}/4\mathbb{Z}$|, where |$\widehat{\alpha }$| is the lift of an automorphism of |$\mathbb{P}^{2}$| of order four |
| 2i | |$\langle (1234),(13) \rangle $| | |$\operatorname{D}_{4}$| | |$\langle \widehat{\alpha } \rangle \simeq \mathbb{Z}/2\mathbb{Z}$|, where |$\widehat{\alpha }$| is the lift of an involution of |$\mathbb{P}^{2}$| |
| 2h | |$\langle (12)(34),(123) \rangle $| | |$\mathcal{A}_{4}$| | |$\lbrace \operatorname{id} \rbrace $| |
| 2j | |$\langle (1234),(12) \rangle $| | |$\operatorname{Sym}_{4}$| | |$\lbrace \operatorname{id} \rbrace $| |
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