Subgroup ($H$) information
| Description: | $C_6^3.(S_3\times D_4)$ |
| Order: | \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
| Index: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(2,18,14)(3,4,17)(5,16,15)(20,22)(21,23), (19,26)(24,25), (4,14)(7,13)(8,11) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_6^4.C_6^2:D_4$ |
| Order: | \(373248\)\(\medspace = 2^{9} \cdot 3^{6} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_6^4.D_6^2.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_3^4.C_2^6.C_2^6$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $(C_3\times C_6^3).C_2.C_2^5$ |
| Normal closure: | $C_3^4.S_4^2:C_2^2$ |
| Core: | $C_3^3:S_3$ |
Other information
| Number of subgroups in this autjugacy class | $72$ |
| Number of conjugacy classes in this autjugacy class | $8$ |
| Möbius function | not computed |
| Projective image | $C_6^4.C_6^2:D_4$ |