Subgroup ($H$) information
| Description: | $A_5^8.D_8$ |
| Order: | \(2687385600000000\)\(\medspace = 2^{20} \cdot 3^{8} \cdot 5^{8} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(13,14,15)(18,19,20)(23,25,24)(36,39,40), (2,4,3)(8,9,10)(12,15,14)(17,19,20) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian and nonsolvable. Whether it is rational has not been computed.
Ambient group ($G$) information
| Description: | $A_5^8.D_4:D_4$ |
| Order: | \(10749542400000000\)\(\medspace = 2^{22} \cdot 3^{8} \cdot 5^{8} \) |
| Exponent: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable. Whether it is rational has not been computed.
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 \(171992678400000000\)\(\medspace = 2^{26} \cdot 3^{8} \cdot 5^{8} \) |
| $\operatorname{Aut}(H)$ | Group of order \(85996339200000000\)\(\medspace = 2^{25} \cdot 3^{8} \cdot 5^{8} \) |
| $\card{W}$ | 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 | $2$ |
| Möbius function | not computed |
| Projective image | not computed |