Subgroup ($H$) information
| Description: | not computed |
| Order: | \(612220032\)\(\medspace = 2^{7} \cdot 3^{14} \) |
| Index: | \(896\)\(\medspace = 2^{7} \cdot 7 \) |
| Exponent: | not computed |
| Generators: |
$\langle(28,30,29)(31,33,32)(34,35,36)(40,41,42), (2,3)(4,6,5)(7,8)(10,12,11)(13,14) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is 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: | $C_3^{14}.C_2^6.C_2^6.C_{14}.C_2$ |
| Order: | \(548549148672\)\(\medspace = 2^{14} \cdot 3^{14} \cdot 7 \) |
| 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: | $C_2^4:F_8$ |
| Order: | \(896\)\(\medspace = 2^{7} \cdot 7 \) |
| Exponent: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Automorphism Group: | $C_2^4:F_8:A_4$, of order \(10752\)\(\medspace = 2^{9} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $A_4$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $2$ |
The quotient is nonabelian, monomial (hence solvable), and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | 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 |