Subgroup ($H$) information
| Description: | $C_{18}$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Index: | \(2189\)\(\medspace = 11 \cdot 199 \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$a^{99}, a^{110}, a^{132}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a Hall subgroup.
Ambient group ($G$) information
| Description: | $C_{199}:C_{198}$ |
| Order: | \(39402\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \cdot 199 \) |
| Exponent: | \(39402\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \cdot 199 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 3$.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{199}:(C_2\times C_{990})$ |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{198}$ | |
| Normalizer: | $C_{198}$ | |
| Normal closure: | $C_{199}:C_{18}$ | |
| Core: | $C_2$ | |
| Minimal over-subgroups: | $C_{199}:C_{18}$ | $C_{198}$ |
| Maximal under-subgroups: | $C_9$ | $C_6$ |
Other information
| Number of subgroups in this conjugacy class | $199$ |
| Möbius function | $1$ |
| Projective image | $C_{199}:C_{99}$ |