Subgroup ($H$) information
| Description: | $S_3^4:C_2$ |
| Order: | \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Index: | \(3\) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(3,5)(4,6)(7,8), (2,5,3)(7,8,9)(10,11,14), (1,6)(2,7)(3,9)(5,8)(10,12,11,15,14,13) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, solvable, and rational. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^4:(D_4\times D_6)$ |
| Order: | \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, solvable, and rational. 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3:S_3\times \He_3).D_4^2$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $\SOPlus(4,2)^2.C_2$, of order \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
| $W$ | $S_3^4:C_2$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $-1$ |
| Projective image | $C_3^4:(D_4\times D_6)$ |