Subgroup ($H$) information
| Description: | $\PSL(3,4):C_2$ |
| Order: | \(40320\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Index: | \(1100\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Generators: |
$\langle(1,38)(2,29)(3,15)(4,98)(5,31)(6,46)(7,12)(8,13)(9,100)(10,68)(11,78)(14,55) \!\cdots\! \rangle$
|
| Derived length: | $1$ |
The subgroup is maximal, nonabelian, almost simple, nonsolvable, and rational.
Ambient group ($G$) information
| Description: | $\HS$ |
| Order: | \(44352000\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5^{3} \cdot 7 \cdot 11 \) |
| Exponent: | \(9240\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \) |
| 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 \(88704000\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5^{3} \cdot 7 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $\PSL(3,4):D_6$, of order \(241920\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5 \cdot 7 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $1100$ |
| Möbius function | not computed |
| Projective image | not computed |