Subgroup ($H$) information
| Description: | $C_7$ |
| Order: | \(7\) |
| Index: | \(38612\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 197 \) |
| Exponent: | \(7\) |
| Generators: |
$b^{197}$
|
| 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, hyperelementary, metacyclic, and a Z-group), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_7\times F_{197}$ |
| Order: | \(270284\)\(\medspace = 2^{2} \cdot 7^{3} \cdot 197 \) |
| Exponent: | \(38612\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 197 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Quotient group ($Q$) structure
| Description: | $F_{197}$ |
| Order: | \(38612\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 197 \) |
| Exponent: | \(38612\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 197 \) |
| Automorphism Group: | $F_{197}$, of order \(38612\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 197 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{1379}.C_7.C_{84}.C_2$ |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $F_{197}$ |