Subgroup ($H$) information
| Description: | $C_{199}$ |
| Order: | \(199\) |
| Index: | \(594\)\(\medspace = 2 \cdot 3^{3} \cdot 11 \) |
| Exponent: | \(199\) |
| Generators: |
$b^{3}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the commutator subgroup (hence characteristic and normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $199$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
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: | $C_3\times C_{198}$ |
| Order: | \(594\)\(\medspace = 2 \cdot 3^{3} \cdot 11 \) |
| Exponent: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| Automorphism Group: | $C_5\times C_6^2:S_3$, of order \(1080\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5 \) |
| Outer Automorphisms: | $C_5\times C_6^2:S_3$, of order \(1080\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (hence hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{199}:(C_{11}:(C_{18}\times S_3))$ |
| $\operatorname{Aut}(H)$ | $C_{198}$, of order \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| $W$ | $C_{198}$, of order \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_3\times F_{199}$ |