Subgroup ($H$) information
| Description: | $C_3^4:C_6$ |
| Order: | \(486\)\(\medspace = 2 \cdot 3^{5} \) |
| Index: | \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\langle(1,31,18)(2,33,17,3,32,16)(4,25,19)(5,27,20,6,26,21)(7,29,13,8,28,15)(9,30,14) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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 \(314928\)\(\medspace = 2^{4} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $C_3^4.C_3^4.C_2^3$, of order \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| $W$ | $C_3^4:(C_3\times C_6)$, of order \(1458\)\(\medspace = 2 \cdot 3^{6} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $72$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $-1$ |
| Projective image | $C_3^6:(C_3\times \SL(2,3))$ |