Subgroup ($H$) information
| Description: | not computed |
| Order: | \(19131876\)\(\medspace = 2^{2} \cdot 3^{14} \) |
| Index: | \(45\)\(\medspace = 3^{2} \cdot 5 \) |
| Exponent: | not computed |
| Generators: |
$\langle(4,5,6)(40,41,42), (4,5)(8,9)(10,12,11)(13,14)(17,18)(23,24)(26,27)(32,33) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group. Whether it is 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 set structure
Since this subgroup has trivial core, the ambient group $G$ acts faithfully and transitively on the set of cosets of $H$. The resulting permutation representation is isomorphic to 45T4982.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | 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 |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $45$ |
| Möbius function | not computed |
| Projective image | not computed |