Subgroup ($H$) information
| Description: | $C_2^{16}.C_6^2.S_3^2$ |
| Order: | \(84934656\)\(\medspace = 2^{20} \cdot 3^{4} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,13,7)(2,14,8)(3,15,9,4,16,10)(5,18,12,6,17,11)(19,30,27)(20,29,28)(21,31,24,22,32,23) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^{16}.C_6^2.\PSU(3,2).C_2^2$ |
| Order: | \(679477248\)\(\medspace = 2^{23} \cdot 3^{4} \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 \(21743271936\)\(\medspace = 2^{28} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | Group of order \(8153726976\)\(\medspace = 2^{25} \cdot 3^{5} \) |
| $\card{W}$ | 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 | $2$ |
| Möbius function | not computed |
| Projective image | not computed |