Subgroup ($H$) information
| Description: | $C_{16}$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(68\)\(\medspace = 2^{2} \cdot 17 \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$ab$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
Ambient group ($G$) information
| Description: | $D_{68}.C_8$ |
| Order: | \(1088\)\(\medspace = 2^{6} \cdot 17 \) |
| Exponent: | \(272\)\(\medspace = 2^{4} \cdot 17 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{17}:((C_2^2\times C_8).C_2^4)$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\operatorname{res}(S)$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(64\)\(\medspace = 2^{6} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{16}$ | |
| Normalizer: | $\OD_{32}$ | |
| Normal closure: | $C_{17}:\OD_{32}$ | |
| Core: | $C_4$ | |
| Minimal over-subgroups: | $C_{17}:C_{16}$ | $\OD_{32}$ |
| Maximal under-subgroups: | $C_8$ |
Other information
| Number of subgroups in this conjugacy class | $34$ |
| Möbius function | $0$ |
| Projective image | $D_{34}:C_4$ |