Subgroup ($H$) information
| Description: | $C_{83}$ |
| Order: | \(83\) |
| Index: | \(334\)\(\medspace = 2 \cdot 167 \) |
| Exponent: | \(83\) |
| Generators: |
$a^{334}$
|
| 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 $83$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{27722}$ |
| Order: | \(27722\)\(\medspace = 2 \cdot 83 \cdot 167 \) |
| Exponent: | \(27722\)\(\medspace = 2 \cdot 83 \cdot 167 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,83,167$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Quotient group ($Q$) structure
| Description: | $C_{334}$ |
| Order: | \(334\)\(\medspace = 2 \cdot 167 \) |
| Exponent: | \(334\)\(\medspace = 2 \cdot 167 \) |
| Automorphism Group: | $C_{166}$, of order \(166\)\(\medspace = 2 \cdot 83 \) |
| Outer Automorphisms: | $C_{166}$, of order \(166\)\(\medspace = 2 \cdot 83 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,167$), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_{6806}$, of order \(13612\)\(\medspace = 2^{2} \cdot 41 \cdot 83 \) |
| $\operatorname{Aut}(H)$ | $C_{82}$, of order \(82\)\(\medspace = 2 \cdot 41 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{27722}$ | |
| Normalizer: | $C_{27722}$ | |
| Complements: | $C_{334}$ | |
| Minimal over-subgroups: | $C_{13861}$ | $C_{166}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $1$ |
| Projective image | $C_{334}$ |