Subgroup ($H$) information
| Description: | $C_{96}$ |
| Order: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Index: | \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
| Exponent: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Generators: |
$b^{579}, b^{9264}, b^{4632}, b^{6176}, b^{1158}, b^{2316}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_{18528}.C_{192}$ |
| Order: | \(3557376\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 193 \) |
| 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: | $F_{193}$ |
| Order: | \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
| Exponent: | \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
| Automorphism Group: | $F_{193}$, of order \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(113836032\)\(\medspace = 2^{16} \cdot 3^{2} \cdot 193 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_8$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{W}$ | 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 |