Subgroup ($H$) information
| Description: | $C_3^{12}.C_2^8.C_3^4.(C_4\times D_4)$ |
| Order: | \(352638738432\)\(\medspace = 2^{13} \cdot 3^{16} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Generators: |
$\langle(16,17,18)(28,30,29), (1,3)(4,6)(10,11,12)(14,15)(17,18)(25,27,26)(28,30,29) \!\cdots\! \rangle$
|
| Derived length: | $5$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^{12}.C_2^8.C_3^4.C_4:C_4.D_4$ |
| Order: | \(1410554953728\)\(\medspace = 2^{15} \cdot 3^{16} \) |
| Exponent: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Derived length: | $5$ |
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 \(5642219814912\)\(\medspace = 2^{17} \cdot 3^{16} \) |
| $\operatorname{Aut}(H)$ | Group of order \(2821109907456\)\(\medspace = 2^{16} \cdot 3^{16} \) |
| $\card{W}$ | 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 | $2$ |
| Möbius function | not computed |
| Projective image | not computed |