Subgroup ($H$) information
| Description: | $\OD_{32}:C_2$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(3\) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$a, b$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_{24}.D_4$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4\times D_4\times D_6$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $C_4^2:C_2^2$, of order \(64\)\(\medspace = 2^{6} \) |
| $\operatorname{res}(S)$ | $C_4^2:C_2^2$, of order \(64\)\(\medspace = 2^{6} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $C_8$ | ||
| Normalizer: | $\OD_{32}:C_2$ | ||
| Normal closure: | $C_{24}.D_4$ | ||
| Core: | $\OD_{16}:C_2$ | ||
| Minimal over-subgroups: | $C_{24}.D_4$ | ||
| Maximal under-subgroups: | $\OD_{16}:C_2$ | $C_2\times C_{16}$ | $\OD_{32}$ |
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $-1$ |
| Projective image | $C_3:D_4$ |