Subgroup ($H$) information
| Description: | $C_7$ |
| Order: | \(7\) |
| Index: | \(238\)\(\medspace = 2 \cdot 7 \cdot 17 \) |
| Exponent: | \(7\) |
| Generators: |
$ab^{170}$
|
| 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 $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_7\times C_{238}$ |
| Order: | \(1666\)\(\medspace = 2 \cdot 7^{2} \cdot 17 \) |
| Exponent: | \(238\)\(\medspace = 2 \cdot 7 \cdot 17 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 7$ (hence hyperelementary), and metacyclic.
Quotient group ($Q$) structure
| Description: | $C_{238}$ |
| Order: | \(238\)\(\medspace = 2 \cdot 7 \cdot 17 \) |
| Exponent: | \(238\)\(\medspace = 2 \cdot 7 \cdot 17 \) |
| Automorphism Group: | $C_2\times C_{48}$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Outer Automorphisms: | $C_2\times C_{48}$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7,17$), hyperelementary, metacyclic, metabelian, a Z-group, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{16}\times C_6.\SO(3,7)$ |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_{238}$ |