Subgroup ($H$) information
| Description: | $C_9$ |
| Order: | \(9\)\(\medspace = 3^{2} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Generators: |
$c$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
Ambient group ($G$) information
| Description: | $C_3.\He_3$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3:D_6$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_3\times C_9$ |
| Normalizer: | $C_3\times C_9$ |
| Normal closure: | $C_3\times C_9$ |
| Core: | $C_3$ |
| Minimal over-subgroups: | $C_3\times C_9$ |
| Maximal under-subgroups: | $C_3$ |
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $0$ |
| Projective image | $\He_3$ |