Subgroup ($H$) information
| Description: | not computed |
| Order: | \(806736\)\(\medspace = 2^{4} \cdot 3 \cdot 7^{5} \) |
| Index: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | not computed |
| Generators: |
$\langle(8,10,14,12,9,13,11), (22,28,23,25,24,26,27), (1,3,7,5,2,6,4)(8,9,10,13,14,11,12) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is nonabelian and solvable. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_7^5:(C_2^4:A_5)$ |
| Order: | \(16134720\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 7^{5} \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and perfect (hence 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^5:(C_2\wr S_5\times C_3)$, of order \(193616640\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \cdot 7^{5} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_7^5.A_4.D_4$ |
| Normal closure: | $C_7^5:(C_2^4:A_5)$ |
| Core: | $C_7^5$ |
Other information
| Number of subgroups in this autjugacy class | $10$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_7^5:(C_2^4:A_5)$ |