Subgroup ($H$) information
| Description: | $C_{4632}$ |
| Order: | \(4632\)\(\medspace = 2^{3} \cdot 3 \cdot 193 \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(4632\)\(\medspace = 2^{3} \cdot 3 \cdot 193 \) |
| Generators: |
$a^{48}, a^{96}, b^{3}, b^{193}, a^{24}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial) and cyclic (hence abelian, elementary ($p = 2,3,193$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{579}:C_{192}$ |
| 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: | $C_{24}$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Automorphism Group: | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
| Outer Automorphisms: | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{579}.C_{24}.C_4^2.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2^3\times C_{192}$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $W$ | $C_{24}$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_{4632}$ | ||
| Normalizer: | $C_{579}:C_{192}$ | ||
| Minimal over-subgroups: | $C_{24}\times C_{193}:C_3$ | $C_3\times C_{193}:C_{16}$ | |
| Maximal under-subgroups: | $C_{2316}$ | $C_{1544}$ | $C_{24}$ |
Other information
| Möbius function | $0$ |
| Projective image | not computed |