Subgroup ($H$) information
| Description: | not computed |
| Order: | \(3538944\)\(\medspace = 2^{17} \cdot 3^{3} \) |
| Index: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$\langle(1,6,11,4,7,9)(2,5,12,3,8,10)(13,22,20,16,24,18)(14,21,19,15,23,17)(25,36,29,26,35,30) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, and metabelian (hence 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: | $C_2^{10}.A_4^3:A_4.D_4$ |
| Order: | \(169869312\)\(\medspace = 2^{21} \cdot 3^{4} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $C_2\times S_4$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Automorphism Group: | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $3$ |
The quotient is nonabelian, monomial (hence solvable), and rational.
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 \(2717908992\)\(\medspace = 2^{25} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | 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 |