Subgroup ($H$) information
| Description: | $C_3.\He_3$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Generators: |
$a^{2}, c^{7}$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^2.C_3^5.C_3.C_6^2$, of order \(236196\)\(\medspace = 2^{2} \cdot 3^{10} \) |
| $\operatorname{Aut}(H)$ | $C_3^3:D_6$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| $\operatorname{res}(S)$ | $C_3^3:C_6$, of order \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| $W$ | $C_3^2:C_6$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $0$ |
| Projective image | $(C_3\times C_9^2):C_6$ |