Subgroup ($H$) information
| Description: | $C_{10}$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Index: | \(497\)\(\medspace = 7 \cdot 71 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$b^{355}, b^{142}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a direct factor, cyclic (hence elementary ($p = 2,5$), hyperelementary, metacyclic, and a Z-group), and a Hall subgroup.
Ambient group ($G$) information
| Description: | $C_{142}:C_{35}$ |
| Order: | \(4970\)\(\medspace = 2 \cdot 5 \cdot 7 \cdot 71 \) |
| Exponent: | \(4970\)\(\medspace = 2 \cdot 5 \cdot 7 \cdot 71 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 7$.
Quotient group ($Q$) structure
| Description: | $C_{71}:C_7$ |
| Order: | \(497\)\(\medspace = 7 \cdot 71 \) |
| Exponent: | \(497\)\(\medspace = 7 \cdot 71 \) |
| Automorphism Group: | $F_{71}$, of order \(4970\)\(\medspace = 2 \cdot 5 \cdot 7 \cdot 71 \) |
| Outer Automorphisms: | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| 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 $p = 7$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{71}:(C_2\times C_{140})$ |
| $\operatorname{Aut}(H)$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4970\)\(\medspace = 2 \cdot 5 \cdot 7 \cdot 71 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{142}:C_{35}$ | |
| Normalizer: | $C_{142}:C_{35}$ | |
| Complements: | $C_{71}:C_7$ | |
| Minimal over-subgroups: | $C_{710}$ | $C_{70}$ |
| Maximal under-subgroups: | $C_5$ | $C_2$ |
Other information
| Möbius function | $71$ |
| Projective image | $C_{71}:C_7$ |