Subgroup ($H$) information
| Description: | $C_{1158}$ |
| Order: | \(1158\)\(\medspace = 2 \cdot 3 \cdot 193 \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(1158\)\(\medspace = 2 \cdot 3 \cdot 193 \) |
| Generators: |
$b^{2316}, b^{1544}, b^{24}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the socle (hence characteristic and normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,193$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{4632}:C_6$ |
| Order: | \(27792\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 193 \) |
| Exponent: | \(4632\)\(\medspace = 2^{3} \cdot 3 \cdot 193 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_2\times C_{12}$ |
| Order: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| Outer Automorphisms: | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4\times D_5\times A_5$, of order \(1778688\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 193 \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_{192}$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $W$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_{4632}$ | |||
| Normalizer: | $C_{4632}:C_6$ | |||
| Minimal over-subgroups: | $C_{1158}:C_3$ | $C_3\times D_{386}$ | $C_{2316}$ | $C_{193}:C_{12}$ |
| Maximal under-subgroups: | $C_{579}$ | $C_{386}$ | $C_6$ |
Other information
| Möbius function | $0$ |
| Projective image | not computed |