Subgroup ($H$) information
| Description: | not computed |
| Order: | \(2687385600000000\)\(\medspace = 2^{20} \cdot 3^{8} \cdot 5^{8} \) |
| Index: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$\langle(2,4)(3,5)(6,8,9,7)(12,13,15,14)(16,19,18,17,20)(22,25)(23,24)(27,28,29,30) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, and nonsolvable. Whether it is a direct factor, a semidirect factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $A_5^8.C_2^4.C_2^5.A_4.C_2$ |
| Order: | \(2063912140800000000\)\(\medspace = 2^{28} \cdot 3^{9} \cdot 5^{8} \) |
| Exponent: | \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and nonsolvable. Whether it is rational has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^6:A_4$ |
| Order: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2^8.A_4.D_6$ |
| Outer Automorphisms: | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $2$ |
The quotient is nonabelian, monomial (hence solvable), and metabelian.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(8255648563200000000\)\(\medspace = 2^{30} \cdot 3^{9} \cdot 5^{8} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |