Subgroup ($H$) information
| Description: | $C_3^2:\SU(3,2)$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$ab^{6}ef^{2}g, cdf, gh^{2}, b^{3}df^{2}h, b^{4}fh^{2}, fh^{2}, h, b^{6}ef^{2}gh$
|
| Derived length: | $4$ |
The subgroup is maximal, nonabelian, and solvable.
Ambient group ($G$) information
| Description: | $\He_3^2:C_4.S_3$ |
| Order: | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times C_3^3:C_3^2.D_4.C_2^3$, of order \(93312\)\(\medspace = 2^{7} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $(C_3^2\times S_3^2):\SD_{16}$, of order \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| $W$ | $C_3^4:Q_8$, of order \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $36$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $-1$ |
| Projective image | not computed |