Subgroup ($H$) information
| Description: | $C_{34}$ |
| Order: | \(34\)\(\medspace = 2 \cdot 17 \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(34\)\(\medspace = 2 \cdot 17 \) |
| Generators: |
$a, c^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,17$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $\OD_{32}:C_{34}$ |
| Order: | \(1088\)\(\medspace = 2^{6} \cdot 17 \) |
| Exponent: | \(272\)\(\medspace = 2^{4} \cdot 17 \) |
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^2\times C_8).C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_{16}$, of order \(16\)\(\medspace = 2^{4} \) |
| $\operatorname{res}(S)$ | $C_{16}$, of order \(16\)\(\medspace = 2^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2\times C_{136}$ | |
| Normalizer: | $C_2\times C_{136}$ | |
| Normal closure: | $D_4\times C_{17}$ | |
| Core: | $C_{17}$ | |
| Minimal over-subgroups: | $C_2\times C_{34}$ | |
| Maximal under-subgroups: | $C_{17}$ | $C_2$ |
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $\OD_{32}:C_2$ |