Subgroup ($H$) information
| Description: | $C_{86}$ |
| Order: | \(86\)\(\medspace = 2 \cdot 43 \) |
| Index: | \(172\)\(\medspace = 2^{2} \cdot 43 \) |
| Exponent: | \(86\)\(\medspace = 2 \cdot 43 \) |
| Generators: |
$a^{172}, b$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,43$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{43}:C_{344}$ |
| Order: | \(14792\)\(\medspace = 2^{3} \cdot 43^{2} \) |
| Exponent: | \(344\)\(\medspace = 2^{3} \cdot 43 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{172}$ |
| Order: | \(172\)\(\medspace = 2^{2} \cdot 43 \) |
| Exponent: | \(172\)\(\medspace = 2^{2} \cdot 43 \) |
| Automorphism Group: | $C_2\times C_{42}$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_2\times C_{42}$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{43}.C_{21}^2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_{42}$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{43}\times C_{172}$ | |
| Normalizer: | $C_{43}:C_{344}$ | |
| Minimal over-subgroups: | $C_{43}\times C_{86}$ | $C_{172}$ |
| Maximal under-subgroups: | $C_{43}$ | $C_2$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{43}:C_{172}$ |