Subgroup ($H$) information
| Description: | $C_6^4.S_3^2$ |
| Order: | \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(1,2,5,9,11,17)(3,7,4,6,12,18)(8,16)(10,15)(13,14)(19,21,24,26)(20,22,25,23) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_6^4.C_6:S_3^2$ |
| Order: | \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times C_6^4.C_3^5.C_2^2$ |
| $\operatorname{Aut}(H)$ | $(C_2\times C_6^3).C_3^4.C_2^3$ |
| $W$ | $C_6^4.S_3^2$, of order \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2\times C_6^4.S_3^2$ |
| Normal closure: | $C_3^5.\POPlus(4,3)$ |
| Core: | $C_6^4.(C_3\times S_3)$ |
Other information
| Number of subgroups in this autjugacy class | $12$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | not computed |
| Projective image | $C_6^4.C_6:S_3^2$ |