Subgroup ($H$) information
| Description: | $C_{2189}:C_{22}$ |
| Order: | \(48158\)\(\medspace = 2 \cdot 11^{2} \cdot 199 \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(4378\)\(\medspace = 2 \cdot 11 \cdot 199 \) |
| Generators: |
$a^{99}, a^{18}, b^{11}, b^{199}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a Hall subgroup, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
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: | $C_9$ |
| Order: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Automorphism Group: | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{2189}.C_{165}.C_6^2$ |
| $\operatorname{Aut}(H)$ | $C_{2189}.C_{495}.C_2^2$ |
| $W$ | $C_{199}:C_{66}$, of order \(13134\)\(\medspace = 2 \cdot 3 \cdot 11 \cdot 199 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | not computed |