Subgroup ($H$) information
| Description: | $C_3^2:D_9$ |
| Order: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$a^{3}, c^{3}, b, c^{7}, d^{3}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $(C_3\times C_9^2):C_6$ |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3^4\times C_9).C_3^4.C_2^3$, of order \(472392\)\(\medspace = 2^{3} \cdot 3^{10} \) |
| $\operatorname{Aut}(H)$ | $C_3^4.C_3^4:(S_3\times \GL(2,3))$, of order \(1889568\)\(\medspace = 2^{5} \cdot 3^{10} \) |
| $\card{\operatorname{res}(S)}$ | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(9\)\(\medspace = 3^{2} \) |
| $W$ | $(C_3^2\times C_9):C_6$, of order \(486\)\(\medspace = 2 \cdot 3^{5} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $1$ |
| Projective image | $(C_3\times C_9^2):C_6$ |