Subgroup ($H$) information
| Description: | $C_{579}$ |
| Order: | \(579\)\(\medspace = 3 \cdot 193 \) |
| Index: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Exponent: | \(579\)\(\medspace = 3 \cdot 193 \) |
| Generators: |
$b^{6176}, b^{96}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 3,193$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{18528}.C_{48}$ |
| Order: | \(889344\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 193 \) |
| 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: | $C_8\times C_{192}$ |
| Order: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Exponent: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Automorphism Group: | $C_2.C_4^3.C_2^6.C_2$ |
| Outer Automorphisms: | $C_2.C_4^3.C_2^6.C_2$ |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{4632}.C_{24}.C_4^2.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{192}$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $W$ | $C_{48}$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_{18528}$ | ||
| Normalizer: | $C_{18528}.C_{48}$ | ||
| Complements: | $C_8\times C_{192}$ | ||
| Minimal over-subgroups: | $C_{579}:C_3$ | $C_{1158}$ | $C_3\times D_{193}$ |
| Maximal under-subgroups: | $C_{193}$ | $C_3$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_{6176}.C_{48}$ |