Subgroup ($H$) information
| Description: | $C_{250}$ |
| Order: | \(250\)\(\medspace = 2 \cdot 5^{3} \) |
| Index: | \(498\)\(\medspace = 2 \cdot 3 \cdot 83 \) |
| Exponent: | \(250\)\(\medspace = 2 \cdot 5^{3} \) |
| Generators: |
$b^{31125}, b^{42330}, b^{24900}, b^{20916}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_{249}\times D_{250}$ |
| Order: | \(124500\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{3} \cdot 83 \) |
| Exponent: | \(62250\)\(\medspace = 2 \cdot 3 \cdot 5^{3} \cdot 83 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{498}$ |
| Order: | \(498\)\(\medspace = 2 \cdot 3 \cdot 83 \) |
| Exponent: | \(498\)\(\medspace = 2 \cdot 3 \cdot 83 \) |
| Automorphism Group: | $C_2\times C_{82}$, of order \(164\)\(\medspace = 2^{2} \cdot 41 \) |
| Outer Automorphisms: | $C_2\times C_{82}$, of order \(164\)\(\medspace = 2^{2} \cdot 41 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,3,83$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_{82}\times (C_{25}.C_{25}.C_5):C_4$, of order \(4100000\)\(\medspace = 2^{5} \cdot 5^{5} \cdot 41 \) |
| $\operatorname{Aut}(H)$ | $C_{100}$, of order \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |