Subgroup ($H$) information
| Description: | $C_3^8.C_3^6:\GL(2,3)$ |
| Order: | \(229582512\)\(\medspace = 2^{4} \cdot 3^{15} \) |
| Index: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Generators: |
$\langle(31,32,33)(34,35,36), (1,20,11,3,19,10,2,21,12)(4,5,6)(7,35,14)(8,36,15) \!\cdots\! \rangle$
|
| Derived length: | $6$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^8.(C_3^8.C_2^3:\GL(2,3))$ |
| Order: | \(16529940864\)\(\medspace = 2^{7} \cdot 3^{17} \) |
| Exponent: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Derived length: | $6$ |
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 \(99179645184\)\(\medspace = 2^{8} \cdot 3^{18} \) |
| $\operatorname{Aut}(H)$ | Group of order \(1377495072\)\(\medspace = 2^{5} \cdot 3^{16} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $36$ |
| Möbius function | not computed |
| Projective image | not computed |