Subgroup ($H$) information
| Description: | $C_2^{12}.(C_6^4.C_6^2:C_4)$ |
| Order: | \(764411904\)\(\medspace = 2^{20} \cdot 3^{6} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(1,32,27,21,2,31,28,22)(3,35,29,24)(4,36,30,23)(5,33,26,20,6,34,25,19)(7,14) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^{12}.(C_6^4.(C_6:D_6.S_3^2))$ |
| Order: | \(13759414272\)\(\medspace = 2^{21} \cdot 3^{8} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(220150628352\)\(\medspace = 2^{25} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | Group of order \(18345885696\)\(\medspace = 2^{23} \cdot 3^{7} \) |
| $\card{W}$ | 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 | $3$ |
| Möbius function | not computed |
| Projective image | not computed |