Subgroup ($H$) information
| Description: | $C_7^3:S_4$ |
| Order: | \(8232\)\(\medspace = 2^{3} \cdot 3 \cdot 7^{3} \) |
| Index: | \(4198072320\)\(\medspace = 2^{15} \cdot 3^{3} \cdot 5 \cdot 13 \cdot 73 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{llll}\alpha^{4} & \alpha^{3} & 0 & \alpha^{4} \\ \alpha^{6} & \alpha^{4} & 1 & \alpha^{6} \\ 0 & \alpha & \alpha^{3} & \alpha^{2} \\ \alpha^{5} & \alpha^{4} & \alpha^{6} & \alpha^{4} \\ \end{array}\right), \left(\begin{array}{llll}0 & 1 & \alpha^{4} & \alpha \\ \alpha^{4} & \alpha^{5} & \alpha & \alpha^{5} \\ \alpha^{3} & \alpha^{3} & 0 & \alpha^{4} \\ \alpha^{3} & \alpha^{3} & 1 & \alpha^{5} \\ \end{array}\right), \left(\begin{array}{llll}\alpha^{2} & \alpha^{3} & 0 & \alpha^{3} \\ 1 & \alpha^{4} & 0 & \alpha^{4} \\ \alpha^{5} & \alpha^{5} & \alpha & \alpha^{4} \\ \alpha^{4} & \alpha & \alpha^{5} & 0 \\ \end{array}\right)$
|
| Derived length: | $4$ |
The subgroup is maximal, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $\SL(4,8)$ |
| Order: | \(34558531338240\)\(\medspace = 2^{18} \cdot 3^{4} \cdot 5 \cdot 7^{3} \cdot 13 \cdot 73 \) |
| Exponent: | \(1195740\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \cdot 73 \) |
| 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 \(207351188029440\)\(\medspace = 2^{19} \cdot 3^{5} \cdot 5 \cdot 7^{3} \cdot 13 \cdot 73 \) |
| $\operatorname{Aut}(H)$ | $C_7^3:(C_6\times S_4)$, of order \(49392\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7^{3} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_7^3:S_4$ |
| Normal closure: | $\SL(4,8)$ |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $4198072320$ |
| Möbius function | not computed |
| Projective image | not computed |