Subgroup ($H$) information
| Description: | $\PSL(3,4):C_2$ |
| Order: | \(40320\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Index: | \(20160\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Exponent: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Generators: |
$\langle(1,4,7)(2,18,3)(6,15,13)(8,14,11)(9,21,16)(10,20,19)(22,37,29)(23,34,31) \!\cdots\! \rangle$
|
| Derived length: | $1$ |
The subgroup is maximal, nonabelian, almost simple, nonsolvable, and rational.
Ambient group ($G$) information
| Description: | $\PSL(3,4)\wr C_2$ |
| Order: | \(812851200\)\(\medspace = 2^{13} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \) |
| Exponent: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
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 \(9754214400\)\(\medspace = 2^{15} \cdot 3^{5} \cdot 5^{2} \cdot 7^{2} \) |
| $\operatorname{Aut}(H)$ | $\PSL(3,4):D_6$, 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 | $20160$ |
| Möbius function | not computed |
| Projective image | not computed |