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: |
$b^{12352}, b^{192}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 3,193$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_{37056}.C_{24}$ |
| 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
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(56918016\)\(\medspace = 2^{15} \cdot 3^{2} \cdot 193 \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_{192}$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |