Subgroup ($H$) information
| Description: | $C_2\times C_7^3:S_4$ |
| Order: | \(16464\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{3} \) |
| Index: | \(1960\)\(\medspace = 2^{3} \cdot 5 \cdot 7^{2} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(16,19)(17,18)(20,21)(23,26)(24,25)(27,28), (15,21,16,18,17,19,20)(22,27,26,24,25,23,28) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_7^5:C_2\wr A_5$ |
| Order: | \(32269440\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \cdot 7^{5} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^5:(C_2^5:\GL(2,4))$, of order \(96808320\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 7^{5} \) |
| $\operatorname{Aut}(H)$ | $D_7^3:(C_6\times S_3)$, of order \(98784\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7^{3} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $490$ |
| Möbius function | not computed |
| Projective image | not computed |