Subgroup ($H$) information
| Description: | not computed |
| Order: | \(100000000\)\(\medspace = 2^{8} \cdot 5^{8} \) |
| Index: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | not computed |
| Generators: |
$\langle(1,2)(3,5)(6,8)(9,10)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(26,27)(28,30) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, and solvable. 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: | $C_5^7.(C_2^3\times F_5).D_4^2:D_4$ |
| Order: | \(6400000000\)\(\medspace = 2^{14} \cdot 5^{8} \) |
| 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: | $C_2^2\times \SD_{16}$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $(D_4\times C_2^5).S_4$, of order \(6144\)\(\medspace = 2^{11} \cdot 3 \) |
| Outer Automorphisms: | $C_2^5:S_4$, of order \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 \(102400000000\)\(\medspace = 2^{18} \cdot 5^{8} \) |
| $\operatorname{Aut}(H)$ | 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 |