Subgroup ($H$) information
| Description: | $C_3^8.C_3^5:(S_3\times \GL(2,3))$ |
| Order: | \(459165024\)\(\medspace = 2^{5} \cdot 3^{15} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Generators: |
$\langle(1,2,3)(10,12,11)(13,15,14)(22,24,23)(25,26,27)(34,36,35), (4,6,5)(16,18,17) \!\cdots\! \rangle$
|
| Derived length: | $6$ |
The subgroup is maximal, nonabelian, and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^{12}.C_3^2.C_2^4.S_4$ |
| Order: | \(1836660096\)\(\medspace = 2^{7} \cdot 3^{15} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $6$ |
The ambient group is nonabelian and solvable. Whether it is monomial 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 \(11019960576\)\(\medspace = 2^{8} \cdot 3^{16} \) |
| $\operatorname{Aut}(H)$ | Group of order \(4132485216\)\(\medspace = 2^{5} \cdot 3^{17} \) |
| $\card{W}$ | 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 | $4$ |
| Möbius function | not computed |
| Projective image | not computed |