Subgroup ($H$) information
| Description: | $C_2\times C_7^3:S_4$ |
| Order: | \(16464\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{3} \) |
| Index: | \(49\)\(\medspace = 7^{2} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Generators: |
$ac^{3}e^{4}f^{6}, d^{7}f^{5}g^{6}, e, b^{2}def^{5}g^{6}, d^{2}f^{3}, fg^{4}, b^{3}c^{8}d^{12}e^{6}f^{6}g^{4}, c^{7}$
|
| Derived length: | $4$ |
The subgroup is maximal, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_7^5:(C_2\times S_4)$ |
| Order: | \(806736\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{5} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^5.A_4.C_6^2.C_2^2$ |
| $\operatorname{Aut}(H)$ | $D_7^3:(C_6\times S_3)$, 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: | $C_7^5:(C_2\times S_4)$ |
| Core: | $C_7:D_7^2$ |
Other information
| Number of subgroups in this autjugacy class | $49$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_7^5:(C_2\times S_4)$ |