Subgroup ($H$) information
| Description: | $D_7\times C_7^2$ |
| Order: | \(686\)\(\medspace = 2 \cdot 7^{3} \) |
| Index: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$b^{3}d^{12}e^{6}f^{5}, e, c^{2}f^{6}, d^{2}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $D_7\times C_7^3:S_4$ |
| Order: | \(115248\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{4} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^3.(C_7\times A_4).C_6^2.C_2$ |
| $\operatorname{Aut}(H)$ | $C_6.\SO(3,7)\times F_7$ |
| $W$ | $D_4\times D_7$, of order \(112\)\(\medspace = 2^{4} \cdot 7 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $0$ |
| Projective image | $D_7\times C_7^3:S_4$ |