Subgroup ($H$) information
| Description: | $D_7^2:C_2^2$ |
| Order: | \(784\)\(\medspace = 2^{4} \cdot 7^{2} \) |
| Index: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Generators: |
$af^{2}, e^{2}, e^{7}f^{7}, cd^{11}f^{6}, d^{7}e^{4}f^{8}, d^{2}e^{8}$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
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).
Quotient set structure
Since this subgroup has trivial core, the ambient group $G$ acts faithfully and transitively on the set of cosets of $H$. The resulting permutation representation is isomorphic to 42T905.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^3.C_2^4.C_6^2.C_2$ |
| $\operatorname{Aut}(H)$ | $C_7:D_7.C_2.C_6.C_2^3$ |
| $W$ | $D_7^2:C_2^2$, of order \(784\)\(\medspace = 2^{4} \cdot 7^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $21$ |
| Möbius function | $0$ |
| Projective image | $(C_7\times C_{14}^2):S_4$ |