Subgroup ($H$) information
| Description: | $C_{1544}$ |
| Order: | \(1544\)\(\medspace = 2^{3} \cdot 193 \) |
| Index: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(1544\)\(\medspace = 2^{3} \cdot 193 \) |
| Generators: |
$b^{2316}, b^{96}, b^{9264}, b^{4632}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,193$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{18528}.C_{16}$ |
| Order: | \(296448\)\(\medspace = 2^{9} \cdot 3 \cdot 193 \) |
| Exponent: | \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_4\times C_{48}$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Automorphism Group: | $C_2^6.D_4$, of order \(512\)\(\medspace = 2^{9} \) |
| Outer Automorphisms: | $C_2^6.D_4$, of order \(512\)\(\medspace = 2^{9} \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{1544}.C_{24}.C_4^2.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_{192}$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| $W$ | $C_{16}$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
| Centralizer: | $C_{18528}$ | |||
| Normalizer: | $C_{18528}.C_{16}$ | |||
| Minimal over-subgroups: | $C_{4632}$ | $C_8\times D_{193}$ | $C_{3088}$ | $C_{193}:C_{16}$ |
| Maximal under-subgroups: | $C_{772}$ | $C_8$ |
Other information
| Möbius function | $0$ |
| Projective image | not computed |