Subgroup ($H$) information
| Description: | $C_{250}$ |
| Order: | \(250\)\(\medspace = 2 \cdot 5^{3} \) |
| Index: | \(2\) |
| Exponent: | \(250\)\(\medspace = 2 \cdot 5^{3} \) |
| Generators: |
$a^{250}, a^{20}, a^{100}, a^{4}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), maximal, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central.
Ambient group ($G$) information
| Description: | $C_{500}$ |
| Order: | \(500\)\(\medspace = 2^{2} \cdot 5^{3} \) |
| Exponent: | \(500\)\(\medspace = 2^{2} \cdot 5^{3} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| 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), a $p$-group, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_{100}$, of order \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $C_{100}$, of order \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{100}$, of order \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{500}$ | |
| Normalizer: | $C_{500}$ | |
| Minimal over-subgroups: | $C_{500}$ | |
| Maximal under-subgroups: | $C_{125}$ | $C_{50}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_2$ |