Subgroup ($H$) information
| Description: | $C_2^4:F_8$ |
| Order: | \(896\)\(\medspace = 2^{7} \cdot 7 \) |
| Index: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Generators: |
$a^{3}, ef, g, cf, fh, h, b^{9}, defgh$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Hall subgroup, monomial (hence solvable), and metabelian.
Ambient group ($G$) information
| Description: | $F_{64}:C_6$ |
| Order: | \(24192\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 7 \) |
| Exponent: | \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_{64}:C_6$, of order \(24192\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2^4:F_8:A_4$, of order \(10752\)\(\medspace = 2^{9} \cdot 3 \cdot 7 \) |
| $W$ | $C_2^3:F_8:C_6$, of order \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $9$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $F_{64}:C_6$ |