Subgroup ($H$) information
| Description: | not computed |
| Order: | \(655360\)\(\medspace = 2^{17} \cdot 5 \) |
| Index: | \(2000\)\(\medspace = 2^{4} \cdot 5^{3} \) |
| Exponent: | not computed |
| Generators: |
$\langle(35,36)(39,40), (11,12)(13,14)(17,18)(19,20)(23,24)(25,26)(31,32)(33,34) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is normal, nonabelian, metabelian (hence solvable), and an A-group. 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: | $C_2^{16}.C_5^3.C_{10}.C_2^4$ |
| Order: | \(1310720000\)\(\medspace = 2^{21} \cdot 5^{4} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_5\times D_{10}^2$ |
| Order: | \(2000\)\(\medspace = 2^{4} \cdot 5^{3} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_{10}^2.A_4.C_4^3.C_2^2$ |
| Outer Automorphisms: | $C_2\wr D_6.C_4$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \) |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
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 \(377487360000\)\(\medspace = 2^{26} \cdot 3^{2} \cdot 5^{4} \) |
| $\operatorname{Aut}(H)$ | 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 |