Subgroup ($H$) information
| Description: | $C_3^3:D_6$ |
| Order: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Index: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$a^{3}b^{3}de^{8}, e^{6}, d, e^{9}, b^{2}ce^{12}, cd^{2}$
|
| Derived length: | $3$ |
The subgroup is nonabelian and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $C_3^4.S_3^3$ |
| Order: | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_3^4.C_2^2\times S_3$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_3^3:C_3^2.Q_8.D_6$ |
| $W$ | $C_3^3.S_3^2$, of order \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $3$ |
| Projective image | $C_3^4.S_3^3$ |