Subgroup ($H$) information
| Description: | $C_7^5:(C_2^4:A_5)$ |
| Order: | \(16134720\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \cdot 7^{5} \) |
| Index: | \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Generators: |
$\langle(2,7)(3,4)(5,6)(9,14)(10,11)(12,13), (29,30,33,31,32,34,35), (1,21,40)(2,16,41,7,18,42) \!\cdots\! \rangle$
|
| Derived length: | $0$ |
The subgroup is nonabelian and perfect (hence nonsolvable).
Ambient group ($G$) information
| Description: | $C_7^6.C_2^5.A_6.C_6$ |
| Order: | \(8131898880\)\(\medspace = 2^{9} \cdot 3^{3} \cdot 5 \cdot 7^{6} \) |
| Exponent: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(16263797760\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5 \cdot 7^{6} \) |
| $\operatorname{Aut}(H)$ | $C_7^5:(C_2\wr S_5\times C_3)$, of order \(193616640\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \cdot 7^{5} \) |
| $\card{W}$ | 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 | $6$ |
| Möbius function | not computed |
| Projective image | not computed |