Subgroup ($H$) information
| Description: | $C_3^6.C_3:S_3^3$ |
| Order: | \(472392\)\(\medspace = 2^{3} \cdot 3^{10} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\langle(4,30,18,5,28,17,6,29,16)(7,20,33)(8,21,31)(9,19,32)(10,34,24,11,36,22,12,35,23) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $C_3^4.\He_3^2:D_4:D_4$ |
| Order: | \(3779136\)\(\medspace = 2^{6} \cdot 3^{10} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_3^4.(C_3\times D_4^2).C_2$, of order \(22674816\)\(\medspace = 2^{7} \cdot 3^{11} \) |
| $\operatorname{Aut}(H)$ | $C_3^6.C_3^4.C_3^3.C_2^3.C_6.C_2^2$, of order \(306110016\)\(\medspace = 2^{6} \cdot 3^{14} \) |
| $W$ | $C_3^4.\He_3^2:D_4:C_2^2$, of order \(1889568\)\(\medspace = 2^{5} \cdot 3^{10} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^4.\He_3^2:D_4:C_2^2$ |
| Normal closure: | $C_3^4.\He_3^2:D_4:C_2^2$ |
| Core: | $C_3^6.C_3^4.C_2$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^4.\He_3^2:D_4:D_4$ |