Subgroup ($H$) information
| Description: | $S_3^2:D_9^2$ |
| Order: | \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
| Index: | \(3\) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$ac^{6}, c^{6}e^{12}, c^{14}e^{16}, d, e^{6}, b^{3}, c^{9}d^{2}e^{9}f^{2}, f, e^{2}, e^{9}$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $(C_3^3\times S_3^2).S_3^2$ |
| Order: | \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_3^3.C_6^2.C_2^4$, of order \(1259712\)\(\medspace = 2^{6} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $C_3^4.C_3^3.(C_3\times D_4^2)$, of order \(419904\)\(\medspace = 2^{6} \cdot 3^{8} \) |
| $W$ | $S_3^2:D_9^2$, of order \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $(C_3^3\times S_3^2).S_3^2$ |