Subgroup ($H$) information
| Description: | not computed |
| Order: | \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | not computed |
| Generators: |
$\langle(1,14,16)(2,9,18), (20,21), (1,2)(7,13)(8,12)(9,16)(10,11)(14,18), (1,5,2) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is nonabelian and supersolvable (hence solvable and monomial). Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_6^2:S_3^2:S_3^2$ |
| Order: | \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_3^2.C_2^6.C_2^4$ |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $C_3^2:S_3^3$, of order \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $8$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | not computed |
| Projective image | $C_6^2:S_3^2:S_3^2$ |