Subgroup ($H$) information
| Description: | not computed |
| Order: | \(10616832\)\(\medspace = 2^{17} \cdot 3^{4} \) |
| Index: | \(512\)\(\medspace = 2^{9} \) |
| Exponent: | not computed |
| Generators: |
$\langle(19,33,24,20,34,23)(21,30,26)(22,29,25)(27,35,31,28,36,32), (1,2)(5,6)(9,10) \!\cdots\! \rangle$
|
| 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: | $C_2^{16}.C_3^4.C_2.D_4^2.C_2^3$ |
| Order: | \(5435817984\)\(\medspace = 2^{26} \cdot 3^{4} \) |
| Exponent: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $D_4^2.C_2^3$ |
| Order: | \(512\)\(\medspace = 2^{9} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | Group of order \(16384\)\(\medspace = 2^{14} \) |
| Outer Automorphisms: | Group of order 128 |
| Derived length: | $3$ |
The quotient is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary). Whether it is metacyclic, monomial, or rational has not been computed.
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 \(21743271936\)\(\medspace = 2^{28} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | 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 |