Subgroup ($H$) information
| Description: | $C_3$ |
| Order: | \(3\) |
| Index: | \(41500\)\(\medspace = 2^{2} \cdot 5^{3} \cdot 83 \) |
| Exponent: | \(3\) |
| Generators: |
$b^{20750}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $3$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
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: | $C_{83}\times D_{250}$ |
| Order: | \(41500\)\(\medspace = 2^{2} \cdot 5^{3} \cdot 83 \) |
| Exponent: | \(20750\)\(\medspace = 2 \cdot 5^{3} \cdot 83 \) |
| Automorphism Group: | $C_2\times C_{82}\times (C_{25}.C_{25}.C_5):C_4$, of order \(2050000\)\(\medspace = 2^{4} \cdot 5^{5} \cdot 41 \) |
| Outer Automorphisms: | $C_2^2\times C_{2050}$ |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, 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 \(4100000\)\(\medspace = 2^{5} \cdot 5^{5} \cdot 41 \) |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\card{W}$ | 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 |