Subgroup ($H$) information
| Description: | not computed |
| Order: | \(65536\)\(\medspace = 2^{16} \) |
| Index: | \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \) |
| Exponent: | not computed |
| Generators: |
$\langle(5,7,6,8)(13,14)(15,16)(17,18)(19,20)(33,35,34,36), (21,22)(23,24)(29,30) \!\cdots\! \rangle$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
The subgroup is normal, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian. 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: | $C_2^{12}.(C_2^9.A_4^3:A_4)$ |
| Order: | \(43486543872\)\(\medspace = 2^{29} \cdot 3^{4} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial or rational has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2^5.A_4^3:A_4$ |
| Order: | \(663552\)\(\medspace = 2^{13} \cdot 3^{4} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Automorphism Group: | Group of order \(3822059520\)\(\medspace = 2^{20} \cdot 3^{6} \cdot 5 \) |
| Outer Automorphisms: | $C_5^3:C_{20}.Q_8$, of order \(11520\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $4$ |
The quotient is nonabelian and solvable. Whether it is monomial has not been computed.
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 \(44530220924928\)\(\medspace = 2^{39} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | 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 |