Subgroup ($H$) information
| Description: | $C_3^2:D_6\times S_4$ |
| Order: | \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Index: | \(7\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,2)(5,6)(8,12,9)(10,13,11)(14,15,16)(17,18), (4,7)(5,6)(8,13,14)(9,10,16) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, a Hall subgroup, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_3^2:D_6\times \GL(3,2)$ |
| Order: | \(18144\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times \SO(3,7)\times \AGL(2,3)$ |
| $\operatorname{Aut}(H)$ | $S_4\times \AGL(2,3).C_2^2$ |
| $W$ | $C_6^2:D_6$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $14$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $-1$ |
| Projective image | $C_3:S_3\times \GL(3,2)$ |