Subgroup ($H$) information
| Description: | $A_5^8.C_4.C_2^4.C_2^4$ |
| Order: | \(171992678400000000\)\(\medspace = 2^{26} \cdot 3^{8} \cdot 5^{8} \) |
| Index: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Exponent: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(27,30,28), (32,33,34), (28,30,29), (16,18,20), (6,8,7), (6,10,7), (26,29,28) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and nonsolvable. Whether it is rational has not been computed.
Ambient group ($G$) information
| Description: | $A_5^8.C_2^4.C_2^6.\PSL(2,7)$ |
| Order: | \(28894769971200000000\)\(\medspace = 2^{29} \cdot 3^{9} \cdot 5^{8} \cdot 7 \) |
| Exponent: | \(5040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(57789539942400000000\)\(\medspace = 2^{30} \cdot 3^{9} \cdot 5^{8} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | Group of order \(687970713600000000\)\(\medspace = 2^{28} \cdot 3^{8} \cdot 5^{8} \) |
| $\card{W}$ | 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 | $84$ |
| Möbius function | not computed |
| Projective image | not computed |