Subgroup ($H$) information
| Description: | $C_3^6:(C_3\times \GL(2,3))$ |
| Order: | \(104976\)\(\medspace = 2^{4} \cdot 3^{8} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Generators: |
$\langle(1,9,2)(3,5,4)(6,8,7)(10,11,18)(12,13,14)(15,16,17)(19,21,26)(20,22,24) \!\cdots\! \rangle$
|
| Derived length: | $5$ |
The subgroup is maximal, nonabelian, and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^6:(A_4\times \GL(2,3))$ |
| Order: | \(419904\)\(\medspace = 2^{6} \cdot 3^{8} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $5$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.(S_4\times \GL(2,3))$, of order \(839808\)\(\medspace = 2^{7} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | $C_3^6.Q_8.C_3^3.C_2^2$, of order \(629856\)\(\medspace = 2^{5} \cdot 3^{9} \) |
| $W$ | $C_3^6:(C_3\times \GL(2,3))$, of order \(104976\)\(\medspace = 2^{4} \cdot 3^{8} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^6:(A_4\times \GL(2,3))$ |