Subgroup ($H$) information
| Description: | not computed |
| Order: | \(2125764\)\(\medspace = 2^{2} \cdot 3^{12} \) |
| Index: | \(49152\)\(\medspace = 2^{14} \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$\langle(7,8,9)(31,33,32)(34,36,35), (2,3)(4,6)(11,12)(14,15)(20,21)(23,24)(28,30) \!\cdots\! \rangle$
|
| 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: | $C_3^{12}.C_2^6.C_4^2.A_4.(C_2\times D_4)$ |
| Order: | \(104485552128\)\(\medspace = 2^{16} \cdot 3^{13} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Order: | \(49152\)\(\medspace = 2^{14} \cdot 3 \) |
| Exponent: | not computed |
| Automorphism Group: | not computed |
| Outer Automorphisms: | not computed |
| Derived length: | not computed |
Properties have 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 \(417942208512\)\(\medspace = 2^{18} \cdot 3^{13} \) |
| $\operatorname{Aut}(H)$ | 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 |