Subgroup ($H$) information
| Description: | not computed |
| Order: | \(100000000\)\(\medspace = 2^{8} \cdot 5^{8} \) |
| Index: | \(8192\)\(\medspace = 2^{13} \) |
| Exponent: | not computed |
| Generators: |
$\langle(31,32,33,34,35), (21,23,25,22,24)(26,29,27,30,28)(31,35,34,33,32), (11,13) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is normal, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group. Whether it is characteristic, 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: | $C_5^8.C_2^4.C_2^6.C_2^6.C_2^5$ |
| Order: | \(819200000000\)\(\medspace = 2^{21} \cdot 5^{8} \) |
| Exponent: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^{10}.D_4$ |
| Order: | \(8192\)\(\medspace = 2^{13} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | Group of order \(6442450944\)\(\medspace = 2^{31} \cdot 3 \) |
| Outer Automorphisms: | Group of order \(6291456\)\(\medspace = 2^{21} \cdot 3 \) |
| Derived length: | $3$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | not computed |
| $\operatorname{Aut}(H)$ | 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 |