Subgroup ($H$) information
| Description: | $C_2\times C_7^3:S_4$ |
| Order: | \(16464\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{3} \) |
| Index: | \(343\)\(\medspace = 7^{3} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Generators: |
$ad^{12}e^{4}f^{4}g^{3}, d^{7}e^{2}f^{4}g^{5}, ef^{6}g^{4}i, b^{2}cde^{5}h^{4}, d^{2}e^{5}f^{3}gh^{2}i, fg^{2}i^{2}, b^{3}e^{6}f^{6}g^{6}hi^{2}, cd^{8}$
|
| Derived length: | $4$ |
The subgroup is maximal, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_7^3.D_7\wr S_3$ |
| Order: | \(5647152\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{6} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^6.A_4.C_6^2.C_2$ |
| $\operatorname{Aut}(H)$ | $D_7^3:(C_6\times S_3)$, of order \(98784\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7^{3} \) |
| $W$ | $C_7^3:S_4$, of order \(8232\)\(\medspace = 2^{3} \cdot 3 \cdot 7^{3} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2\times C_7^3:S_4$ |
| Normal closure: | $C_7^3.D_7\wr S_3$ |
| Core: | $C_7^3$ |
Other information
| Number of subgroups in this autjugacy class | $343$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_7^3.D_7\wr S_3$ |