Subgroup ($H$) information
| Description: | $D_8\wr C_2$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$a, cde^{2}f^{4}, f$
|
| 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: | $C_4^2:C_2^2.D_4^2$ |
| Order: | \(4096\)\(\medspace = 2^{12} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_2^4.C_2^6.C_2^3$ |
| $\operatorname{Aut}(H)$ | $(C_4\times \OD_{16}).D_4^2$, 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 | $2$ |
| Möbius function | not computed |
| Projective image | not computed |