Subgroup ($H$) information
| Description: | $C_{200}$ |
| Order: | \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
| Index: | \(402\)\(\medspace = 2 \cdot 3 \cdot 67 \) |
| Exponent: | \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
| Generators: |
$a^{50}, a^{100}, a^{80}, a^{16}, a^{200}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and cyclic (hence elementary ($p = 2,5$), hyperelementary, metacyclic, and a Z-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: | $D_{201}$ |
| Order: | \(402\)\(\medspace = 2 \cdot 3 \cdot 67 \) |
| Exponent: | \(402\)\(\medspace = 2 \cdot 3 \cdot 67 \) |
| Automorphism Group: | $C_{67}:(C_{66}\times S_3)$, of order \(26532\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \cdot 67 \) |
| Outer Automorphisms: | $C_{66}$, of order \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 \(4245120\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 67 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_{20}$, of order \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{201}:C_{400}$ | ||
| Normalizer: | $C_{201}:C_{400}$ | ||
| Minimal over-subgroups: | $C_{13400}$ | $C_{600}$ | $C_{400}$ |
| Maximal under-subgroups: | $C_{100}$ | $C_{40}$ |
Other information
| Möbius function | $-201$ |
| Projective image | $D_{201}$ |