Subgroup ($H$) information
| Description: | $C_7:Q_{16}$ |
| Order: | \(112\)\(\medspace = 2^{4} \cdot 7 \) |
| Index: | \(5\) |
| Exponent: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Generators: |
$a, b^{6}, b^{4}, b, c^{5}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Hall subgroup, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{70}.(C_2^4\times C_{12})$ |
| $\operatorname{Aut}(H)$ | $C_2\times D_4\times F_7$, of order \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_2\times D_4\times F_7$, of order \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_7:D_4$, of order \(56\)\(\medspace = 2^{3} \cdot 7 \) |
Related subgroups
| Centralizer: | $C_2$ | |||
| Normalizer: | $C_7:Q_{16}$ | |||
| Normal closure: | $C_{35}:Q_{16}$ | |||
| Core: | $C_7\times Q_8$ | |||
| Minimal over-subgroups: | $C_{35}:Q_{16}$ | |||
| Maximal under-subgroups: | $C_7\times Q_8$ | $C_7:Q_8$ | $C_7:C_8$ | $Q_{16}$ |
Other information
| Number of subgroups in this conjugacy class | $5$ |
| Möbius function | $-1$ |
| Projective image | $C_{35}:D_4$ |