Subgroup ($H$) information
| Description: | not computed |
| Order: | \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | not computed |
| Generators: |
$\langle(1,33,16,2,32,18)(3,31,17)(4,26,21,5,25,19)(6,27,20)(7,28,15)(8,30,14,9,29,13) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is maximal, nonabelian, and solvable. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^6:(C_3\times \SL(2,3))$ |
| Order: | \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_2.C_6^2.C_6$, of order \(314928\)\(\medspace = 2^{4} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $9$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_3^6:(C_3\times \SL(2,3))$ |