Subgroup ($H$) information
| Description: | $A_8$ |
| Order: | \(20160\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Index: | \(5040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Generators: |
$\langle(8,15)(9,11)(10,12)(13,14), (9,15,14,11,10)\rangle$
|
| Derived length: | $0$ |
The subgroup is characteristic (hence normal), nonabelian, and simple (hence nonsolvable, perfect, quasisimple, and almost simple). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $A_7.S_8$ |
| Order: | \(101606400\)\(\medspace = 2^{10} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \) |
| Exponent: | \(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_7\times S_8$, of order \(203212800\)\(\medspace = 2^{11} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2} \) |
| $\operatorname{Aut}(H)$ | $S_8$, of order \(40320\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $\card{W}$ | 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 |