Subgroup ($H$) information
| Description: | not computed |
| Order: | \(531441\)\(\medspace = 3^{12} \) |
| Index: | \(1620\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 5 \) |
| Exponent: | not computed |
| Generators: |
$\langle(4,6,5)(13,15,14)(22,24,23)(31,33,32)(40,41,42), (16,18,17)(28,29,30)(37,38,39) \!\cdots\! \rangle$
|
| Nilpotency class: | not computed |
| Derived length: | not computed |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), and a $p$-group (hence elementary and hyperelementary). 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: | $C_5\times C_3^3:A_4$ |
| Order: | \(1620\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 5 \) |
| Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Automorphism Group: | $C_4\times S_3\wr S_3$, of order \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| Outer Automorphisms: | $C_2^2\times C_4$, of order \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $3$ |
The quotient is nonabelian and monomial (hence solvable).
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 |