Subgroup ($H$) information
| Description: | $C_3^4.(C_3^8.C_6\wr S_4)$ |
| Order: | \(16529940864\)\(\medspace = 2^{7} \cdot 3^{17} \) |
| Index: | \(15400\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 7 \cdot 11 \) |
| Exponent: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Generators: |
$\langle(4,6,5)(7,8,9)(10,12,11)(13,14,15)(16,17,18)(34,36,35), (25,26,27)(34,35,36) \!\cdots\! \rangle$
|
| Derived length: | $6$ |
The subgroup is maximal, nonabelian, and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^{11}.A_{12}.C_6$ |
| Order: | \(254561089305600\)\(\medspace = 2^{10} \cdot 3^{17} \cdot 5^{2} \cdot 7 \cdot 11 \) |
| Exponent: | \(83160\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 11 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
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 \(509122178611200\)\(\medspace = 2^{11} \cdot 3^{17} \cdot 5^{2} \cdot 7 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | Group of order \(99179645184\)\(\medspace = 2^{8} \cdot 3^{18} \) |
| $\card{W}$ | 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 | $15400$ |
| Möbius function | not computed |
| Projective image | not computed |