Subgroup ($H$) information
| Description: | not computed |
| Order: | \(2744\)\(\medspace = 2^{3} \cdot 7^{3} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$ad^{7}f^{6}, d^{2}e^{6}, e^{2}, f^{2}, f^{7}, e^{7}f^{7}$
|
| Derived length: | not computed |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian. Whether it is elementary, metacyclic, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $(C_7\times C_{14}^2):S_4$ |
| Order: | \(32928\)\(\medspace = 2^{5} \cdot 3 \cdot 7^{3} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^3.C_2^4.C_6^2.C_2$ |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $C_2\times D_{14}$, of order \(56\)\(\medspace = 2^{3} \cdot 7 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $6$ |
| Möbius function | $2$ |
| Projective image | $(C_7\times C_{14}^2):S_4$ |