Subgroup ($H$) information
| Description: | $\He_3:S_4$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Index: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(4,5)(6,7)(8,13,9,10,12,11)(14,15,16)(17,18), (2,7,4)(3,5,6)(8,11,16)(9,13,15) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times \SO(3,7)\times \AGL(2,3)$ |
| $\operatorname{Aut}(H)$ | $S_4\times C_3^2:\GL(2,3)$, of order \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
| $W$ | $C_6^2:D_6$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $28$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $C_6:S_3\times \PSL(2,7)$ |