Subgroup ($H$) information
| Description: | $C_{4150}$ |
| Order: | \(4150\)\(\medspace = 2 \cdot 5^{2} \cdot 83 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4150\)\(\medspace = 2 \cdot 5^{2} \cdot 83 \) |
| Generators: |
$b^{8300}, b^{11288}, b^{6640}, b^{200}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5,83$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{16600}:C_2$ |
| Order: | \(33200\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 83 \) |
| Exponent: | \(16600\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 83 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Quotient group ($Q$) structure
| Description: | $D_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{50}.C_{410}.C_2^5$, of order \(656000\)\(\medspace = 2^{7} \cdot 5^{3} \cdot 41 \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_{820}$, of order \(1640\)\(\medspace = 2^{3} \cdot 5 \cdot 41 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{16600}$ | ||
| Normalizer: | $C_{16600}:C_2$ | ||
| Minimal over-subgroups: | $C_{8300}$ | $C_{83}\times D_{50}$ | $C_{25}:C_{332}$ |
| Maximal under-subgroups: | $C_{2075}$ | $C_{830}$ | $C_{50}$ |
Other information
| Möbius function | $0$ |
| Projective image | $D_{100}$ |