Subgroup ($H$) information
| Description: | $C_{579}$ |
| Order: | \(579\)\(\medspace = 3 \cdot 193 \) |
| Index: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(579\)\(\medspace = 3 \cdot 193 \) |
| Generators: |
$b^{193}, b^{3}$
|
| 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 ($p = 3,193$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_3\times F_{193}$ |
| Order: | \(111168\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 193 \) |
| Exponent: | \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{192}$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Automorphism Group: | $C_2^2\times C_{16}$, of order \(64\)\(\medspace = 2^{6} \) |
| Outer Automorphisms: | $C_2^2\times C_{16}$, of order \(64\)\(\medspace = 2^{6} \) |
| Nilpotency class: | $1$ |
| 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{579}.C_{96}.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{192}$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $W$ | $C_{192}$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_{579}$ | |
| Normalizer: | $C_3\times F_{193}$ | |
| Complements: | $C_{192}$ $C_{192}$ $C_{192}$ | |
| Minimal over-subgroups: | $C_{579}:C_3$ | $C_3\times D_{193}$ |
| Maximal under-subgroups: | $C_{193}$ | $C_3$ |
Other information
| Möbius function | $0$ |
| Projective image | $F_{193}$ |