Subgroup ($H$) information
| Description: | $C_2^5.\SD_{16}$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$af, cefg$
|
| Nilpotency class: | $6$ |
| Derived length: | $3$ |
The subgroup is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary).
Ambient group ($G$) information
| Description: | $C_2^6.C_2\wr C_4$ |
| Order: | \(4096\)\(\medspace = 2^{12} \) |
| 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).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^8.C_2^6.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2^5.(D_4\times C_2^4)$, of order \(4096\)\(\medspace = 2^{12} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| 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 | $4$ |
| Möbius function | not computed |
| Projective image | not computed |