Subgroup ($H$) information
| Description: | $C_{199}:C_{11}$ |
| Order: | \(2189\)\(\medspace = 11 \cdot 199 \) |
| Index: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(2189\)\(\medspace = 11 \cdot 199 \) |
| Generators: |
$a^{18}b^{10}, b^{11}$
|
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 11$.
Ambient group ($G$) information
| Description: | $C_{2189}:C_{198}$ |
| Order: | \(433422\)\(\medspace = 2 \cdot 3^{2} \cdot 11^{2} \cdot 199 \) |
| 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.
Quotient group ($Q$) structure
| Description: | $C_{198}$ |
| Order: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| Automorphism Group: | $C_2\times C_{30}$, of order \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Outer Automorphisms: | $C_2\times C_{30}$, of order \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,11$), hyperelementary, metacyclic, metabelian, a Z-group, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{2189}.C_{165}.C_6^2$ |
| $\operatorname{Aut}(H)$ | $F_{199}$, of order \(39402\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \cdot 199 \) |
| $W$ | $C_{199}:C_{66}$, of order \(13134\)\(\medspace = 2 \cdot 3 \cdot 11 \cdot 199 \) |
Related subgroups
| Centralizer: | $C_{33}$ | ||
| Normalizer: | $C_{2189}:C_{198}$ | ||
| Complements: | $C_{198}$ $C_{198}$ | ||
| Minimal over-subgroups: | $C_{2189}:C_{11}$ | $C_{199}:C_{33}$ | $C_{199}:C_{22}$ |
| Maximal under-subgroups: | $C_{199}$ | $C_{11}$ |
Other information
| Number of subgroups in this autjugacy class | $11$ |
| Number of conjugacy classes in this autjugacy class | $11$ |
| Möbius function | $0$ |
| Projective image | $C_{2189}:C_{198}$ |