Subgroup ($H$) information
| Description: | $C_3^6.C_3^3:(C_4\times S_3)$ |
| Order: | \(472392\)\(\medspace = 2^{3} \cdot 3^{10} \) |
| Index: | \(3\) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(4,5,6)(13,14,15)(19,20,21)(22,23,24)(25,27,26)(28,30,29)(31,33,32)(34,36,35) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^6.C_3^4:(C_4\times S_3)$ |
| Order: | \(1417176\)\(\medspace = 2^{3} \cdot 3^{11} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(459165024\)\(\medspace = 2^{5} \cdot 3^{15} \) |
| $\operatorname{Aut}(H)$ | Group of order \(153055008\)\(\medspace = 2^{5} \cdot 3^{14} \) |
| $W$ | $C_3^6.C_3^3:(C_4\times S_3)$, of order \(472392\)\(\medspace = 2^{3} \cdot 3^{10} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^6.C_3^3:(C_4\times S_3)$ |
| Normal closure: | $C_3^6.C_3^4:(C_4\times S_3)$ |
| Core: | $C_3^6.C_3^3.D_6$ |
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^6.C_3^4:(C_4\times S_3)$ |