Subgroup ($H$) information
| Description: | $C_{1379}$ |
| Order: | \(1379\)\(\medspace = 7 \cdot 197 \) |
| Index: | \(196\)\(\medspace = 2^{2} \cdot 7^{2} \) |
| Exponent: | \(1379\)\(\medspace = 7 \cdot 197 \) |
| Generators: |
$b^{197}, b^{7}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Fitting subgroup (hence characteristic, normal, nilpotent, solvable, supersolvable, and monomial), the socle, a semidirect factor, and cyclic (hence abelian, elementary ($p = 7,197$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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: | $C_{196}$ |
| Order: | \(196\)\(\medspace = 2^{2} \cdot 7^{2} \) |
| Exponent: | \(196\)\(\medspace = 2^{2} \cdot 7^{2} \) |
| Automorphism Group: | $C_2\times C_{42}$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_2\times C_{42}$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7$), hyperelementary, metacyclic, metabelian, a Z-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{1379}.C_7.C_{84}.C_2$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{588}$, of order \(1176\)\(\medspace = 2^{3} \cdot 3 \cdot 7^{2} \) |
| $W$ | $C_{196}$, of order \(196\)\(\medspace = 2^{2} \cdot 7^{2} \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $F_{197}$ |