Subgroup ($H$) information
| Description: | $D_{17}$ |
| Order: | \(34\)\(\medspace = 2 \cdot 17 \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(34\)\(\medspace = 2 \cdot 17 \) |
| Generators: |
$a, c^{4}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times D_4\times F_{17}$, of order \(4352\)\(\medspace = 2^{8} \cdot 17 \) |
| $\operatorname{Aut}(H)$ | $F_{17}$, of order \(272\)\(\medspace = 2^{4} \cdot 17 \) |
| $\operatorname{res}(S)$ | $F_{17}$, of order \(272\)\(\medspace = 2^{4} \cdot 17 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_{17}:C_4$, of order \(68\)\(\medspace = 2^{2} \cdot 17 \) |
Related subgroups
| Centralizer: | $C_4$ | |
| Normalizer: | $D_{17}:C_8$ | |
| Normal closure: | $D_{68}$ | |
| Core: | $C_{17}$ | |
| Minimal over-subgroups: | $D_{34}$ | |
| Maximal under-subgroups: | $C_{17}$ | $C_2$ |
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $D_{68}.C_8$ |