Subgroup ($H$) information
| Description: | $C_5^6.(D_5\times S_7)$ |
| Order: | \(787500000\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5^{8} \cdot 7 \) |
| Index: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(2100\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \) |
| Generators: |
$\langle(16,17,18,19,20)(21,22,23,24,25)(26,29,27,30,28)(31,33,35,32,34)(36,38,40,37,39) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian and nonsolvable.
Ambient group ($G$) information
| Description: | $C_5^8.A_9.C_2^2$ |
| Order: | \(283500000000\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 5^{9} \cdot 7 \) |
| Exponent: | \(12600\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
| 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 \(567000000000\)\(\medspace = 2^{9} \cdot 3^{4} \cdot 5^{9} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | Group of order \(6300000000\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5^{8} \cdot 7 \) |
| $\card{W}$ | 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 | $180$ |
| Möbius function | not computed |
| Projective image | not computed |