Subgroup ($H$) information
| Description: | not computed |
| Order: | \(25000000\)\(\medspace = 2^{6} \cdot 5^{8} \) |
| Index: | \(512\)\(\medspace = 2^{9} \) |
| Exponent: | not computed |
| Generators: |
$\langle(2,4,5,3)(7,9,10,8)(12,14,15,13)(17,19,20,18)(22,24,25,23)(27,29,30,28) \!\cdots\! \rangle$
|
| 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: | $C_5^8.C_4^3.C_2^4.C_4^2.C_2$ |
| Order: | \(12800000000\)\(\medspace = 2^{15} \cdot 5^{8} \) |
| Exponent: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_8^2:(C_2\times C_4)$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Automorphism Group: | $(C_2^2\times D_4^2).C_2^5$, of order \(8192\)\(\medspace = 2^{13} \) |
| Outer Automorphisms: | $C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 \(102400000000\)\(\medspace = 2^{18} \cdot 5^{8} \) |
| $\operatorname{Aut}(H)$ | 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 |