Subgroup ($H$) information
| Description: | $C_3^2.S_3^3$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(1,18,15)(2,17,16)(3,12)(4,7)(5,14,9)(6,10)(8,11)(19,22)(20,21), (1,17,16,5,14,15) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and rational.
Ambient group ($G$) information
| Description: | $C_6^2:S_3^2:S_3^2$ |
| Order: | \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_3^2.C_2^6.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_3^4:(S_3\times D_4)$, of order \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| $W$ | $C_3^2.S_3^3$, of order \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $96$ |
| Number of conjugacy classes in this autjugacy class | $16$ |
| Möbius function | not computed |
| Projective image | $C_6^2:S_3^2:S_3^2$ |