Subgroup ($H$) information
| Description: | $C_3^4.D_6^2:D_6$ |
| Order: | \(139968\)\(\medspace = 2^{6} \cdot 3^{7} \) |
| Index: | \(3\) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(19,21)(20,22), (1,16,17)(2,4,10,11,9,18,8,15,13)(3,5,6)(7,12,14), (1,9,17,2,5,18) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^4.D_6^2:S_3^2$ |
| Order: | \(419904\)\(\medspace = 2^{6} \cdot 3^{8} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and rational. 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_3^2.C_2^4.C_6.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_3^6.C_6.C_2.C_6.C_2^6$ |
| $W$ | $C_3^6.(D_4\times D_6)$, of order \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^4.D_6^2:D_6$ |
| Normal closure: | $C_3^4.D_6^2:S_3^2$ |
| Core: | $C_3^4.D_6^2:S_3$ |
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_3^5:S_3.D_6^2$ |