Subgroup ($H$) information
| Description: | $C_6^4.S_3^2$ |
| Order: | \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \) |
| Index: | \(4096\)\(\medspace = 2^{12} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(19,20)(21,22)(23,24)(25,29,27,26,30,28) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^{12}.(C_6^4.S_3^2)$ |
| Order: | \(191102976\)\(\medspace = 2^{18} \cdot 3^{6} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^{12}.C_6^2.C_6^2.C_3^2.C_2^3$ |
| $\operatorname{Aut}(H)$ | $(C_2\times C_6^3).C_3^4.C_2^3$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_6^4.S_3^2$ |
| Normal closure: | $C_2^{12}.(C_6^4.S_3^2)$ |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $4096$ |
| Möbius function | not computed |
| Projective image | not computed |