Subgroup ($H$) information
| Description: | $(C_3^2\times C_9):C_6$ |
| Order: | \(486\)\(\medspace = 2 \cdot 3^{5} \) |
| Index: | \(3\) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$a^{3}, e^{3}, d, a^{2}, bc^{2}e^{8}, cd^{2}e^{3}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $(C_3^3\times C_9):C_6$ |
| Order: | \(1458\)\(\medspace = 2 \cdot 3^{6} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_3^4.D_6$, of order \(708588\)\(\medspace = 2^{2} \cdot 3^{11} \) |
| $\operatorname{Aut}(H)$ | $C_3^5.S_3^3$, of order \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| $\card{\operatorname{res}(S)}$ | \(26244\)\(\medspace = 2^{2} \cdot 3^{8} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(9\)\(\medspace = 3^{2} \) |
| $W$ | $(C_3^2\times C_9):C_6$, of order \(486\)\(\medspace = 2 \cdot 3^{5} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $(C_3^3\times C_9):C_6$ |