Subgroup ($H$) information
| Description: | $C_{10050}$ |
| Order: | \(10050\)\(\medspace = 2 \cdot 3 \cdot 5^{2} \cdot 67 \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(10050\)\(\medspace = 2 \cdot 3 \cdot 5^{2} \cdot 67 \) |
| Generators: |
$b^{3}, a^{80}, a^{16}, b^{67}, a^{200}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,5,67$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{201}:C_{400}$ |
| Order: | \(80400\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 67 \) |
| Exponent: | \(80400\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 67 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Quotient group ($Q$) structure
| Description: | $C_8$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{201}.C_{330}.C_2.C_2^5$, of order \(4245120\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 67 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_{660}$, of order \(2640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{40200}$ | |||
| Normalizer: | $C_{201}:C_{400}$ | |||
| Minimal over-subgroups: | $C_{20100}$ | |||
| Maximal under-subgroups: | $C_{5025}$ | $C_{3350}$ | $C_{2010}$ | $C_{150}$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{201}:C_8$ |