Subgroup ($H$) information
| Description: | $C_3^{12}.(C_5\times A_4)$ |
| Order: | \(31886460\)\(\medspace = 2^{2} \cdot 3^{13} \cdot 5 \) |
| Index: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Generators: |
$\langle(28,30,29)(31,32,33)(34,35,36)(37,39,38)(40,42,41)(43,45,44), (16,18,17) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and solvable. Whether it is monomial 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.
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 \(220399211520\)\(\medspace = 2^{10} \cdot 3^{16} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | Group of order \(4081466880\)\(\medspace = 2^{9} \cdot 3^{13} \cdot 5 \) |
| $\card{W}$ | 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 | $27$ |
| Möbius function | not computed |
| Projective image | not computed |