Subgroup ($H$) information
| Description: | $C_{32}$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
| Exponent: | \(32\)\(\medspace = 2^{5} \) |
| Generators: |
$b^{193}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a direct factor, cyclic (hence elementary, hyperelementary, metacyclic, and a Z-group), and a $p$-group.
Ambient group ($G$) information
| Description: | $C_{32}\times F_{193}$ |
| Order: | \(1185792\)\(\medspace = 2^{11} \cdot 3 \cdot 193 \) |
| Exponent: | \(37056\)\(\medspace = 2^{6} \cdot 3 \cdot 193 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{3088}.C_{24}.C_8^2.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_8$, of order \(16\)\(\medspace = 2^{4} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{32}\times F_{193}$ | ||
| Normalizer: | $C_{32}\times F_{193}$ | ||
| Complements: | $F_{193}$ | ||
| Minimal over-subgroups: | $C_{6176}$ | $C_{96}$ | $C_2\times C_{32}$ |
| Maximal under-subgroups: | $C_{16}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $F_{193}$ |