Subgroup ($H$) information
| Description: | $C_7:C_8$ |
| Order: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Generators: |
$b, b^{4}, c^{5}, b^{2}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{35}:Q_{16}$ |
| Order: | \(560\)\(\medspace = 2^{4} \cdot 5 \cdot 7 \) |
| Exponent: | \(280\)\(\medspace = 2^{3} \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{70}.(C_2^4\times C_{12})$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times F_7$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_2^2\times F_7$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $D_{14}$, of order \(28\)\(\medspace = 2^{2} \cdot 7 \) |
Related subgroups
| Centralizer: | $C_4$ | |
| Normalizer: | $C_7:Q_{16}$ | |
| Normal closure: | $C_{35}:C_8$ | |
| Core: | $C_{28}$ | |
| Minimal over-subgroups: | $C_{35}:C_8$ | $C_7:Q_{16}$ |
| Maximal under-subgroups: | $C_{28}$ | $C_8$ |
Other information
| Number of subgroups in this conjugacy class | $5$ |
| Möbius function | $1$ |
| Projective image | $C_{35}:D_4$ |