Subgroup ($H$) information
| Description: | $C_3^{12}.C_2^8.C_3^4.C_4^2.C_2^2$ |
| Order: | \(705277476864\)\(\medspace = 2^{14} \cdot 3^{16} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Generators: |
$\langle(16,17,18)(20,21)(22,23,24)(25,27,26)(31,33), (14,15)(26,27)(31,32,33), (14,15) \!\cdots\! \rangle$
|
| Derived length: | $5$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^{12}.C_2^8.C_3^4.C_4^2.C_2^4.C_6$ |
| Order: | \(16926659444736\)\(\medspace = 2^{17} \cdot 3^{17} \) |
| 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 \(67706637778944\)\(\medspace = 2^{19} \cdot 3^{17} \) |
| $\operatorname{Aut}(H)$ | Group of order \(5642219814912\)\(\medspace = 2^{17} \cdot 3^{16} \) |
| $\card{W}$ | 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 | $6$ |
| Möbius function | not computed |
| Projective image | not computed |