Subgroup ($H$) information
| Description: | $D_9$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Index: | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$a, b^{704}, b^{528}$
|
| 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: | $D_{792}$ |
| Order: | \(1584\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(792\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{99}.C_{60}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_9:C_6$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| $\operatorname{res}(S)$ | $C_9:C_6$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $W$ | $D_9$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_2$ | |
| Normalizer: | $D_{18}$ | |
| Normal closure: | $D_{396}$ | |
| Core: | $C_9$ | |
| Minimal over-subgroups: | $D_{99}$ | $D_{18}$ |
| Maximal under-subgroups: | $C_9$ | $S_3$ |
| Autjugate subgroups: | 1584.67.88.b1.b1 |
Other information
| Number of subgroups in this conjugacy class | $44$ |
| Möbius function | $0$ |
| Projective image | $D_{792}$ |