Subgroup ($H$) information
| Description: | $C_2^4:F_8$ |
| Order: | \(896\)\(\medspace = 2^{7} \cdot 7 \) |
| Index: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Generators: |
$\langle(2,5)(3,12)(6,24)(8,14)(9,20)(11,18)(15,19)(21,22), (2,8)(3,19)(5,14)(6,20) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), and metabelian.
Ambient group ($G$) information
| Description: | $C_2^6.(D_6\times F_8)$ |
| Order: | \(43008\)\(\medspace = 2^{11} \cdot 3 \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^5.\SOPlus(4,2)$, of order \(903168\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 7^{2} \) |
| $\operatorname{Aut}(H)$ | $C_2^4:F_8:A_4$, of order \(10752\)\(\medspace = 2^{9} \cdot 3 \cdot 7 \) |
| $\card{W}$ | 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 | $24$ |
| Möbius function | not computed |
| Projective image | not computed |