Subgroup ($H$) information
| Description: | $C_{199}:C_9$ |
| Order: | \(1791\)\(\medspace = 3^{2} \cdot 199 \) |
| Index: | \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Exponent: | \(1791\)\(\medspace = 3^{2} \cdot 199 \) |
| Generators: |
$a^{110}b^{2}, a^{132}b^{204}, b^{3}$
|
| 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 $p = 3$.
Ambient group ($G$) information
| Description: | $C_3\times F_{199}$ |
| Order: | \(118206\)\(\medspace = 2 \cdot 3^{3} \cdot 11 \cdot 199 \) |
| 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: | $C_{66}$ |
| Order: | \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Exponent: | \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Automorphism Group: | $C_2\times C_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Outer Automorphisms: | $C_2\times C_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{199}:(C_{11}:(C_{18}\times S_3))$ |
| $\operatorname{Aut}(H)$ | $F_{199}$, 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
| Möbius function | $-1$ |
| Projective image | $C_3\times F_{199}$ |