Subgroup ($H$) information
| Description: | $C_{597}$ |
| Order: | \(597\)\(\medspace = 3 \cdot 199 \) |
| Index: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(597\)\(\medspace = 3 \cdot 199 \) |
| Generators: |
$b^{199}, b^{3}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the socle, a semidirect factor, and cyclic (hence abelian, elementary ($p = 3,199$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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: | $C_{198}$ |
| Order: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| Exponent: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| Automorphism Group: | $C_2\times C_{30}$, of order \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Outer Automorphisms: | $C_2\times C_{30}$, of order \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Nilpotency class: | $1$ |
| 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{199}:(C_{11}:(C_{18}\times S_3))$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{198}$, of order \(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \) |
| $W$ | $C_{198}$, of order \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $F_{199}$ |