Subgroup ($H$) information
| Description: | $C_{150}$ |
| Order: | \(150\)\(\medspace = 2 \cdot 3 \cdot 5^{2} \) |
| Index: | \(536\)\(\medspace = 2^{3} \cdot 67 \) |
| Exponent: | \(150\)\(\medspace = 2 \cdot 3 \cdot 5^{2} \) |
| Generators: |
$a^{200}, a^{16}, a^{80}, b^{67}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,5$), 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: | $C_{67}:C_8$ |
| Order: | \(536\)\(\medspace = 2^{3} \cdot 67 \) |
| Exponent: | \(536\)\(\medspace = 2^{3} \cdot 67 \) |
| Automorphism Group: | $C_{67}:(C_2^2\times C_{66})$, of order \(17688\)\(\medspace = 2^{3} \cdot 3 \cdot 11 \cdot 67 \) |
| Outer Automorphisms: | $C_2\times C_{66}$, of order \(132\)\(\medspace = 2^{2} \cdot 3 \cdot 11 \) |
| 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 \(4245120\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 67 \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_{20}$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{40200}$ | ||
| Normalizer: | $C_{201}:C_{400}$ | ||
| Minimal over-subgroups: | $C_{10050}$ | $C_{300}$ | |
| Maximal under-subgroups: | $C_{75}$ | $C_{50}$ | $C_{30}$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_{201}:C_8$ |