Subgroup ($H$) information
| Description: | not computed |
| Order: | \(71744535\)\(\medspace = 3^{15} \cdot 5 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$\langle(19,20,21)(25,27,26)(28,29,30)(34,36,35)(40,42,41)(43,45,44), (43,45,44) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is characteristic (hence normal), nonabelian, metabelian (hence solvable), 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: | $C_3^{15}.(C_5\times A_4)$ |
| Order: | \(860934420\)\(\medspace = 2^{2} \cdot 3^{16} \cdot 5 \) |
| Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| 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: | $A_4$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $2$ |
The quotient is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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 \(220399211520\)\(\medspace = 2^{10} \cdot 3^{16} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | 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 |