Subgroup ($H$) information
| Description: | $C_{35}$ |
| Order: | \(35\)\(\medspace = 5 \cdot 7 \) |
| Index: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Exponent: | \(35\)\(\medspace = 5 \cdot 7 \) |
| Generators: |
$c^{21}, bc^{15}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a semidirect factor, and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 5,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_7:D_{35}$ |
| Order: | \(490\)\(\medspace = 2 \cdot 5 \cdot 7^{2} \) |
| Exponent: | \(70\)\(\medspace = 2 \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $D_7$ |
| Order: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Automorphism Group: | $F_7$, of order \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_3$, of order \(3\) |
| 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 $p = 2$.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_5\times C_7^2:C_3.\SL(2,7).C_2$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{12}$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_2\times C_{12}$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(10290\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7^{3} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Möbius function | $7$ |
| Projective image | $C_7:D_{35}$ |