Subgroup ($H$) information
| Description: | $C_2^6:S_4$ |
| Order: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Index: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(17,18)(19,21)(20,22)(23,24), (1,5)(2,9)(3,10)(4,12)(6,14)(7,11)(8,15)(13,16) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian, monomial (hence solvable), and rational.
Ambient group ($G$) information
| Description: | $C_2^7:\PSL(2,7)$ |
| Order: | \(21504\)\(\medspace = 2^{10} \cdot 3 \cdot 7 \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and perfect (hence 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^7.\PSL(2,7)$ |
| $\operatorname{Aut}(H)$ | $C_2^6.S_4^2$, of order \(36864\)\(\medspace = 2^{12} \cdot 3^{2} \) |
| $W$ | $C_2^5:S_4$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $7$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_2^6:\GL(3,2)$ |