Subgroup ($H$) information
| Description: | $C_3\times C_{66}$ |
| Order: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| Index: | \(597\)\(\medspace = 3 \cdot 199 \) |
| Exponent: | \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Generators: |
$a^{99}, b^{199}, a^{18}, a^{132}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_3\times F_{199}$ |
| Order: | \(118206\)\(\medspace = 2 \cdot 3^{3} \cdot 11 \cdot 199 \) |
| Exponent: | \(39402\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \cdot 199 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{199}:(C_{11}:(C_{18}\times S_3))$ |
| $\operatorname{Aut}(H)$ | $C_{10}\times \GL(2,3)$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $199$ |
| Möbius function | $1$ |
| Projective image | $F_{199}$ |