Subgroup ($H$) information
| Description: | $C_7:C_{28}$ |
| Order: | \(196\)\(\medspace = 2^{2} \cdot 7^{2} \) |
| Index: | \(7\) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Generators: |
$a, a^{2}, b, c$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $\He_7:C_4$ |
| Order: | \(1372\)\(\medspace = 2^{2} \cdot 7^{3} \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_7\times C_{14}):\GL(2,7)$, of order \(197568\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 7^{3} \) |
| $\operatorname{Aut}(H)$ | $C_{14}:C_6^2$, of order \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
| $\operatorname{res}(S)$ | $C_{14}:C_6^2$, of order \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(7\) |
| $W$ | $D_7$, of order \(14\)\(\medspace = 2 \cdot 7 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $7$ |
| Möbius function | $-1$ |
| Projective image | $C_7:D_7$ |