Subgroup ($H$) information
| Description: | $\OD_{32}$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$a, c^{9}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $D_4.F_9$ |
| Order: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^2\times F_9).C_2^3$ |
| $\operatorname{Aut}(H)$ | $D_4:C_2^2$, of order \(32\)\(\medspace = 2^{5} \) |
| $\operatorname{res}(S)$ | $D_4:C_2$, of order \(16\)\(\medspace = 2^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $C_8$ | |
| Normalizer: | $\OD_{32}:C_2$ | |
| Normal closure: | $C_3^2:\OD_{32}$ | |
| Core: | $C_4$ | |
| Minimal over-subgroups: | $C_3^2:\OD_{32}$ | $\OD_{32}:C_2$ |
| Maximal under-subgroups: | $C_2\times C_8$ | $C_{16}$ |
Other information
| Number of subgroups in this conjugacy class | $9$ |
| Möbius function | $1$ |
| Projective image | $C_6^2:C_8$ |