Subgroup ($H$) information
| Description: | $C_2\times C_7^3:S_4$ |
| Order: | \(16464\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{3} \) |
| Index: | \(2205\)\(\medspace = 3^{2} \cdot 5 \cdot 7^{2} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(2,7)(3,4)(5,6)(9,14)(10,11)(12,13)(15,28,17,26,21,25,19,27,20,23,16,24,18,22) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_7^5:S_6.C_3$ |
| Order: | \(36303120\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5 \cdot 7^{5} \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^5:(C_6\times S_6)$, of order \(72606240\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \cdot 7^{5} \) |
| $\operatorname{Aut}(H)$ | $D_7^3:(C_6\times S_3)$, of order \(98784\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7^{3} \) |
| $W$ | $C_7^3:(C_3\times S_4)$, of order \(24696\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7^{3} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_7^3:(C_6\times S_4)$ |
| Normal closure: | $C_7^5:S_6$ |
| Core: | $C_1$ |
Other information
| Number of subgroups in this autjugacy class | $735$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_7^5:S_6.C_3$ |