Subgroup ($H$) information
| Description: | $C_7:D_7$ |
| Order: | \(98\)\(\medspace = 2 \cdot 7^{2} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$abc, c^{8}, b^{2}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_{56}.D_{14}$ |
| Order: | \(1568\)\(\medspace = 2^{5} \cdot 7^{2} \) |
| Exponent: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^2.C_6^2.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_7^2.\GL(2,7)$, of order \(98784\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7^{3} \) |
| $\operatorname{res}(S)$ | $F_7^2$, of order \(1764\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $D_7^2$, of order \(196\)\(\medspace = 2^{2} \cdot 7^{2} \) |
Related subgroups
| Centralizer: | $C_2$ | |||||
| Normalizer: | $C_{14}.D_{14}$ | |||||
| Normal closure: | $C_7:D_{28}$ | |||||
| Core: | $C_7^2$ | |||||
| Minimal over-subgroups: | $C_7:D_{14}$ | |||||
| Maximal under-subgroups: | $C_7^2$ | $D_7$ | $D_7$ | $D_7$ | $D_7$ | $D_7$ |
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $C_{56}.D_{14}$ |