Subgroup ($H$) information
| Description: | not computed |
| Order: | \(39402\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \cdot 199 \) |
| Index: | \(11\) |
| Exponent: | not computed |
| Generators: |
$a^{132}, b^{11}, a^{18}, a^{110}, a^{99}$
|
| Derived length: | not computed |
The subgroup is normal, maximal, a direct factor, nonabelian, and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group). Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
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: | $C_{11}$ |
| Order: | \(11\) |
| Exponent: | \(11\) |
| Automorphism Group: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| Outer Automorphisms: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{2189}.C_{165}.C_6^2$ |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $C_{199}:C_{66}$, of order \(13134\)\(\medspace = 2 \cdot 3 \cdot 11 \cdot 199 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $11$ |
| Number of conjugacy classes in this autjugacy class | $11$ |
| Möbius function | $-1$ |
| Projective image | not computed |