Subgroup ($H$) information
| Description: | $C_6^3:D_4$ |
| Order: | \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(12,19,16)(14,15,18), (21,24)(22,23), (9,11,10), (21,23)(22,24), (1,6)(3,7) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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 \(165888\)\(\medspace = 2^{11} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $\ASL(2,3).C_2^5.C_2^5.C_2^3.\PSL(2,7)$ |
| $W$ | $D_6:D_6$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
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 |