Subgroup ($H$) information
| Description: | $C_2\times C_{198}$ |
| Order: | \(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \) |
| Index: | \(1791\)\(\medspace = 3^{2} \cdot 199 \) |
| Exponent: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| Generators: |
$a^{99}b^{2880}, a^{110}b^{16}, a^{18}b^{2718}, b^{1791}, a^{132}b^{1632}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_{18}\times F_{199}$ |
| Order: | \(709236\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 11 \cdot 199 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{1791}.C_{33}.C_6^2.C_2$ |
| $\operatorname{Aut}(H)$ | $D_6\times C_{30}$, of order \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $1791$ |
| Number of conjugacy classes in this autjugacy class | $9$ |
| Möbius function | $0$ |
| Projective image | $C_9\times F_{199}$ |