Subgroup ($H$) information
| Description: | $\OD_{32}:C_2$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$a, b^{4}c^{2}$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_4^2.C_2\wr C_4$ |
| Order: | \(1024\)\(\medspace = 2^{10} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $7$ |
| Derived length: | $2$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4^3.C_2^4.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2^3.C_2^4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\card{W}$ | \(32\)\(\medspace = 2^{5} \) |
Related subgroups
| Centralizer: | $C_4$ | |
| Normalizer: | $C_4^2.C_8$ | |
| Normal closure: | $C_4^2.C_2^2:C_8$ | |
| Core: | $C_2^2\times C_4$ | |
| Minimal over-subgroups: | $C_4^2.C_8$ | |
| Maximal under-subgroups: | $C_2\times \OD_{16}$ | $\OD_{32}$ |
Other information
| Number of subgroups in this conjugacy class | $8$ |
| Möbius function | $0$ |
| Projective image | not computed |