Subgroup ($H$) information
| Description: | $C_{250}$ |
| Order: | \(250\)\(\medspace = 2 \cdot 5^{3} \) |
| Index: | \(5\) |
| Exponent: | \(250\)\(\medspace = 2 \cdot 5^{3} \) |
| Generators: |
$b^{125}, b^{210}, b^{50}, ab^{6}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, maximal, a semidirect factor, and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{125}:C_{10}$ |
| Order: | \(1250\)\(\medspace = 2 \cdot 5^{4} \) |
| Exponent: | \(250\)\(\medspace = 2 \cdot 5^{3} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 5$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).
Quotient group ($Q$) structure
| Description: | $C_5$ |
| Order: | \(5\) |
| Exponent: | \(5\) |
| Automorphism Group: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| 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, and simple.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(\He_5.C_5):C_4$, of order \(2500\)\(\medspace = 2^{2} \cdot 5^{4} \) |
| $\operatorname{Aut}(H)$ | $C_{100}$, of order \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| $\operatorname{res}(S)$ | $C_{100}$, of order \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(5\) |
| $W$ | $C_5$, of order \(5\) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_5^2$ |