Subgroup ($H$) information
| Description: | $\Sp(4,8)$ |
| Order: | \(1056706560\)\(\medspace = 2^{12} \cdot 3^{4} \cdot 5 \cdot 7^{2} \cdot 13 \) |
| Index: | \(32832\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 19 \) |
| Exponent: | \(16380\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \) |
| Generators: |
$\left(\begin{array}{llll}\alpha^{57} & \alpha^{33} & \alpha & \alpha^{9} \\ \alpha^{22} & \alpha^{4} & \alpha^{54} & \alpha^{8} \\ \alpha^{29} & \alpha^{27} & \alpha^{57} & \alpha^{12} \\ \alpha^{45} & \alpha^{43} & \alpha^{50} & \alpha^{4} \\ \end{array}\right), \left(\begin{array}{llll}\alpha^{6} & \alpha^{31} & \alpha^{26} & \alpha^{37} \\ \alpha^{11} & \alpha^{53} & \alpha^{35} & 0 \\ \alpha^{30} & \alpha^{56} & \alpha^{58} & \alpha^{9} \\ \alpha^{62} & \alpha & \alpha^{4} & \alpha^{46} \\ \end{array}\right)$
|
| Derived length: | $0$ |
The subgroup is maximal, nonabelian, and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
Ambient group ($G$) information
| Description: | $\SU(4,8)$ |
| Order: | \(34693789777920\)\(\medspace = 2^{18} \cdot 3^{7} \cdot 5 \cdot 7^{2} \cdot 13 \cdot 19 \) |
| Exponent: | \(933660\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 13 \cdot 19 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(208162738667520\)\(\medspace = 2^{19} \cdot 3^{8} \cdot 5 \cdot 7^{2} \cdot 13 \cdot 19 \) |
| $\operatorname{Aut}(H)$ | Group of order \(6340239360\)\(\medspace = 2^{13} \cdot 3^{5} \cdot 5 \cdot 7^{2} \cdot 13 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $\Sp(4,8)$ |
| Normal closure: | $\SU(4,8)$ |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $32832$ |
| Möbius function | not computed |
| Projective image | not computed |