Subgroup ($H$) information
| Description: | not computed |
| Order: | \(2799360000000\)\(\medspace = 2^{14} \cdot 3^{7} \cdot 5^{7} \) |
| Index: | \(5040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Exponent: | not computed |
| Generators: |
$\langle(2,4)(3,5)(6,10,9)(11,15,14)(17,20,18)(21,24,23)(26,27,30,29,28)(32,35,34) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, an A-group, and perfect (hence 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: | $A_5^7.A_7.C_2$ |
| Order: | \(14108774400000000\)\(\medspace = 2^{18} \cdot 3^{9} \cdot 5^{8} \cdot 7 \) |
| Exponent: | \(12600\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $S_7$ |
| Order: | \(5040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Automorphism Group: | $S_7$, of order \(5040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is nonabelian, almost simple, nonsolvable, and rational.
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 \(28217548800000000\)\(\medspace = 2^{19} \cdot 3^{9} \cdot 5^{8} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | 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 |