Subgroup ($H$) information
| Description: | not computed |
| Order: | \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \) |
| Index: | \(3\) |
| Exponent: | not computed |
| Generators: |
$\langle(12,19,16)(14,15,18), (21,24)(22,23), (9,11,10), (21,23)(22,24), (2,5)(4,8) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is maximal, nonabelian, supersolvable (hence solvable and monomial), and metabelian. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $(C_2^3\times C_6^3):D_6$ |
| Order: | \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_7^3:C_6$, of order \(165888\)\(\medspace = 2^{11} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $C_2^4:S_3^2$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |