Subgroup ($H$) information
| Description: | $C_3^3:S_4$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Index: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(2,5)(3,7,6,4,8,9)(10,18)(11,16)(14,17)(20,21), (1,5)(4,7,9)(6,8)(19,20) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $(C_3^3\times C_6^2):S_4$ |
| Order: | \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_2^4.D_6^2$ |
| $\operatorname{Aut}(H)$ | $S_3\wr S_3$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| $W$ | $C_3^3:S_4$, of order \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $72$ |
| Number of conjugacy classes in this autjugacy class | $6$ |
| Möbius function | $0$ |
| Projective image | $(C_3^3\times C_6^2):S_4$ |