Subgroup ($H$) information
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(332\)\(\medspace = 2^{2} \cdot 83 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$ab^{166}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a $p$-group.
Ambient group ($G$) information
| Description: | $\OD_{16}\times C_{83}$ |
| Order: | \(1328\)\(\medspace = 2^{4} \cdot 83 \) |
| Exponent: | \(664\)\(\medspace = 2^{3} \cdot 83 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).
Quotient group ($Q$) structure
| Description: | $C_{332}$ |
| Order: | \(332\)\(\medspace = 2^{2} \cdot 83 \) |
| Exponent: | \(332\)\(\medspace = 2^{2} \cdot 83 \) |
| Automorphism Group: | $C_2\times C_{82}$, of order \(164\)\(\medspace = 2^{2} \cdot 41 \) |
| Outer Automorphisms: | $C_2\times C_{82}$, of order \(164\)\(\medspace = 2^{2} \cdot 41 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,83$), 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4:C_{82}$, of order \(1312\)\(\medspace = 2^{5} \cdot 41 \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(656\)\(\medspace = 2^{4} \cdot 41 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_2\times C_{332}$ | |
| Normalizer: | $\OD_{16}\times C_{83}$ | |
| Minimal over-subgroups: | $C_{332}$ | $C_2\times C_4$ |
| Maximal under-subgroups: | $C_2$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_2\times C_{332}$ |