Subgroup ($H$) information
| Description: | $C_2^5.\SD_{16}$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\langle(1,31,3,29)(2,25,5,23)(4,24,6,26)(7,16,13,12)(8,11,14,15)(9,32,10,30)(17,19,22,28) \!\cdots\! \rangle$
|
| Nilpotency class: | $6$ |
| Derived length: | $3$ |
The subgroup is normal, nonabelian, and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_2^5.C_8:C_2^5$ |
| Order: | \(8192\)\(\medspace = 2^{13} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Nilpotency class: | $6$ |
| Derived length: | $3$ |
The ambient group is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary).
Quotient group ($Q$) structure
| Description: | $C_2^4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $A_8$, of order \(20160\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Outer Automorphisms: | $A_8$, of order \(20160\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and rational.
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 \(402653184\)\(\medspace = 2^{27} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_2^5.(D_4\times C_2^4)$, of order \(4096\)\(\medspace = 2^{12} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |