Subgroup ($H$) information
| Description: | not computed |
| Order: | \(234375\)\(\medspace = 3 \cdot 5^{7} \) |
| Index: | \(450\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | not computed |
| Generators: |
$\langle(31,35,34,33,32)(36,40,39,38,37), (1,45,10)(2,44,9)(3,43,8)(4,42,7)(5,41,6) \!\cdots\! \rangle$
|
| 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: | $C_5^6.C_{15}^2:D_{15}$ |
| Order: | \(105468750\)\(\medspace = 2 \cdot 3^{3} \cdot 5^{9} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| 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: | $C_3\times C_5^2:S_3$ |
| Order: | \(450\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Automorphism Group: | $C_5^2:(C_4\times D_6)$, of order \(1200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \) |
| Outer Automorphisms: | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Derived length: | $3$ |
The quotient is nonabelian, monomial (hence solvable), 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 \(13500000000\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5^{9} \) |
| $\operatorname{Aut}(H)$ | 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 |