Subgroup ($H$) information
| Description: | $F_{199}$ |
| Order: | \(39402\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \cdot 199 \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(39402\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \cdot 199 \) |
| Generators: |
$a^{132}b^{612}, b^{18}, a^{18}, a^{110}b^{6}, a^{99}$
|
| Derived length: | $2$ |
The subgroup is normal, a direct factor, nonabelian, and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).
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.
Quotient group ($Q$) structure
| Description: | $C_{18}$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| 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 ($p = 2,3$), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{1791}.C_{33}.C_6^2.C_2$ |
| $\operatorname{Aut}(H)$ | $F_{199}$, of order \(39402\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \cdot 199 \) |
| $W$ | $F_{199}$, of order \(39402\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \cdot 199 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $18$ |
| Number of conjugacy classes in this autjugacy class | $18$ |
| Möbius function | $0$ |
| Projective image | $C_{18}\times F_{199}$ |