Subgroup ($H$) information
| Description: | $C_3^2:D_6\times S_4$ |
| Order: | \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Index: | \(104\)\(\medspace = 2^{3} \cdot 13 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(15,17)(18,19), (1,6,4)(2,10,8)(9,12,13)(14,18)(15,17)(16,19), (14,16)(18,19) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $S_4\times \GL(3,3)$ |
| Order: | \(269568\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 13 \) |
| Exponent: | \(312\)\(\medspace = 2^{3} \cdot 3 \cdot 13 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times \SL(3,3).C_2\times S_4$ |
| $\operatorname{Aut}(H)$ | $S_4\times \AGL(2,3).C_2^2$ |
| $W$ | $S_4\times S_3^2$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $52$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | not computed |