Subgroup ($H$) information
| Description: | $C_2^8$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Index: | \(896\)\(\medspace = 2^{7} \cdot 7 \) |
| Exponent: | \(2\) |
| Generators: |
$\langle(3,4)(5,6)(7,8)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(27,28) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $\ASigmaL(1,16384)$ |
| Order: | \(229376\)\(\medspace = 2^{15} \cdot 7 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and metabelian (hence solvable). Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^4:F_8$ |
| Order: | \(896\)\(\medspace = 2^{7} \cdot 7 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Automorphism Group: | $C_2^4:F_8:A_4$, of order \(10752\)\(\medspace = 2^{9} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $A_4$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, monomial (hence solvable), and metabelian.
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 \(308281344\)\(\medspace = 2^{21} \cdot 3 \cdot 7^{2} \) |
| $\operatorname{Aut}(H)$ | $\GL(8,2)$ |
| $W$ | $C_{14}$, of order \(14\)\(\medspace = 2 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_2^{14}$ |
| Normalizer: | $\ASigmaL(1,16384)$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_2\times C_2^{12}.C_{14}$ |