Subgroup ($H$) information
| Description: | $C_3^4:(C_3\times C_6)$ |
| Order: | \(1458\)\(\medspace = 2 \cdot 3^{6} \) |
| Index: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\langle(10,11,12)(13,15,14)(16,18,17)(19,20,21)(22,24,23)(25,27,26)(28,30,29)(31,33,32) \!\cdots\! \rangle$
|
| 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 \(314928\)\(\medspace = 2^{4} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $C_3^3.C_3^4.D_4.C_2$, of order \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| $W$ | $C_3^4:(C_3\times C_6)$, of order \(1458\)\(\medspace = 2 \cdot 3^{6} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $36$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_3^6:(C_3\times \SL(2,3))$ |