Subgroup ($H$) information
| Description: | not computed |
| Order: | \(44079842304\)\(\medspace = 2^{10} \cdot 3^{16} \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | not computed |
| Generators: |
$\langle(23,24)(35,36), (19,21,20), (2,3)(13,15,14)(22,23,24)(25,27), (5,6)(7,33,19,8,31,20,9,32,21) \!\cdots\! \rangle$
|
| 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: | $C_3^{12}.C_2^8.C_3^4.C_2.C_2\wr C_2^2$ |
| Order: | \(1410554953728\)\(\medspace = 2^{15} \cdot 3^{16} \) |
| Exponent: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2\times D_8$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_4.D_4^2$, of order \(256\)\(\medspace = 2^{8} \) |
| Outer Automorphisms: | $C_2^2\wr C_2$, of order \(32\)\(\medspace = 2^{5} \) |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 \(5642219814912\)\(\medspace = 2^{17} \cdot 3^{16} \) |
| $\operatorname{Aut}(H)$ | 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 |