Subgroup ($H$) information
| Description: | $D_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(4116\)\(\medspace = 2^{2} \cdot 3 \cdot 7^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$af^{2}, d^{7}f^{8}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^3.C_2^4.C_6^2.C_2$ |
| $\operatorname{Aut}(H)$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $2058$ |
| Möbius function | $-2$ |
| Projective image | $(C_7\times C_{14}^2):S_4$ |