Subgroup ($H$) information
| Description: | $C_3\times \He_3$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(3\) |
| Generators: |
$a, c, e$
|
| Nilpotency class: | $2$ |
| 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: | $C_3^3:\He_3$ |
| Order: | \(729\)\(\medspace = 3^{6} \) |
| Exponent: | \(3\) |
| Nilpotency class: | $2$ |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^9.C_2.\SL(3,3)$, of order \(221079456\)\(\medspace = 2^{5} \cdot 3^{12} \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_3^4:(S_3\times \GL(2,3))$, of order \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| $\operatorname{res}(S)$ | $C_3^3.S_3^3$, of order \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(81\)\(\medspace = 3^{4} \) |
| $W$ | $C_3^2$, of order \(9\)\(\medspace = 3^{2} \) |
Related subgroups
| Centralizer: | $C_3^3$ | ||
| Normalizer: | $C_3^2\times \He_3$ | ||
| Normal closure: | $C_3^2\times \He_3$ | ||
| Core: | $C_3^3$ | ||
| Minimal over-subgroups: | $C_3^2\times \He_3$ | ||
| Maximal under-subgroups: | $C_3^3$ | $\He_3$ | $C_3^3$ |
Other information
| Number of subgroups in this autjugacy class | $468$ |
| Number of conjugacy classes in this autjugacy class | $156$ |
| Möbius function | $0$ |
| Projective image | $C_3\times \He_3$ |