Subgroup ($H$) information
| Description: | $C_{83}\times D_{100}$ |
| Order: | \(16600\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 83 \) |
| Index: | $1$ |
| Exponent: | \(8300\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 83 \) |
| Generators: |
$b^{4150}, b^{664}, b^{3320}, b^{100}, b^{2075}, a$
|
| Derived length: | $2$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, metacyclic (hence supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{83}\times D_{100}$ |
| Order: | \(16600\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 83 \) |
| Exponent: | \(8300\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 83 \) |
| 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: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{50}.C_{410}.C_2^4$, of order \(328000\)\(\medspace = 2^{6} \cdot 5^{3} \cdot 41 \) |
| $\operatorname{Aut}(H)$ | $C_{50}.C_{410}.C_2^4$, of order \(328000\)\(\medspace = 2^{6} \cdot 5^{3} \cdot 41 \) |
| $W$ | $D_{50}$, of order \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
Related subgroups
| Centralizer: | $C_{166}$ | ||||
| Normalizer: | $C_{83}\times D_{100}$ | ||||
| Complements: | $C_1$ | ||||
| Maximal under-subgroups: | $C_{83}\times D_{50}$ | $C_{83}\times D_{50}$ | $C_{8300}$ | $D_{20}\times C_{83}$ | $D_{100}$ |
Other information
| Möbius function | $1$ |
| Projective image | $D_{50}$ |