Subgroup ($H$) information
| Description: | not computed |
| Order: | \(4782969\)\(\medspace = 3^{14} \) |
| Index: | \(229376\)\(\medspace = 2^{15} \cdot 7 \) |
| Exponent: | not computed |
| Generators: |
$\langle(19,21,20)(22,23,24)(31,32,33)(34,35,36)(40,42,41), (28,30,29)(34,36,35) \!\cdots\! \rangle$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
The subgroup is characteristic (hence normal), a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $3$-Sylow subgroup (hence a Hall subgroup), and a $p$-group (hence elementary and hyperelementary). Whether it is a direct factor, elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^{14}.C_2^4.C_2^6.F_8.C_2^2$ |
| Order: | \(1097098297344\)\(\medspace = 2^{15} \cdot 3^{14} \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial or rational has not been computed.
Quotient group ($Q$) structure
| Description: | $\ASigmaL(1,16384)$ |
| Order: | \(229376\)\(\medspace = 2^{15} \cdot 7 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Automorphism Group: | Group of order \(308281344\)\(\medspace = 2^{21} \cdot 3 \cdot 7^{2} \) |
| Outer Automorphisms: | Group of order \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian and metabelian (hence solvable). Whether it is monomial 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 \(6582589784064\)\(\medspace = 2^{16} \cdot 3^{15} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | 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 |