Subgroup ($H$) information
| Description: | not computed |
| Order: | \(12500000\)\(\medspace = 2^{5} \cdot 5^{8} \) |
| Index: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | not computed |
| Generators: |
$\langle(21,24,22,25,23)(31,34,32,35,33)(36,39,37,40,38), (2,4,5,3)(7,9,10,8)(12,14,15,13) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, supersolvable (hence solvable and monomial), metabelian, 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: | $C_5^7.(C_2^3\times F_5).C_2^6:C_4$ |
| Order: | \(3200000000\)\(\medspace = 2^{13} \cdot 5^{8} \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^4.(C_2\times D_4)$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2^7.D_4^2$, of order \(8192\)\(\medspace = 2^{13} \) |
| Outer Automorphisms: | $C_2\times D_4^2$, of order \(128\)\(\medspace = 2^{7} \) |
| 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 \(51200000000\)\(\medspace = 2^{17} \cdot 5^{8} \) |
| $\operatorname{Aut}(H)$ | 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 |