Subgroup ($H$) information
| Description: | $C_{19}$ |
| Order: | \(19\) |
| Index: | \(344\)\(\medspace = 2^{3} \cdot 43 \) |
| Exponent: | \(19\) |
| Generators: |
$a^{1376}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $19$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{6536}$ |
| Order: | \(6536\)\(\medspace = 2^{3} \cdot 19 \cdot 43 \) |
| Exponent: | \(6536\)\(\medspace = 2^{3} \cdot 19 \cdot 43 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,19,43$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Quotient group ($Q$) structure
| Description: | $C_{344}$ |
| Order: | \(344\)\(\medspace = 2^{3} \cdot 43 \) |
| Exponent: | \(344\)\(\medspace = 2^{3} \cdot 43 \) |
| Automorphism Group: | $C_2^2\times C_{42}$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_2^2\times C_{42}$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,43$), hyperelementary, metacyclic, metabelian, a Z-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_6\times C_{126}$ |
| $\operatorname{Aut}(H)$ | $C_{18}$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{6536}$ | |
| Normalizer: | $C_{6536}$ | |
| Complements: | $C_{344}$ | |
| Minimal over-subgroups: | $C_{817}$ | $C_{38}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{344}$ |