Subgroup ($H$) information
| Description: | $C_3.\He_3$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Generators: |
$ae, bc$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_3^2.C_3^4$ |
| Order: | \(729\)\(\medspace = 3^{6} \) |
| 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.
Quotient group ($Q$) structure
| Description: | $C_3^2$ |
| Order: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(3\) |
| Automorphism Group: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Outer Automorphisms: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3\times C_3^4.\He_3.D_6$, of order \(78732\)\(\medspace = 2^{2} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $C_3^3:D_6$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| $\operatorname{res}(S)$ | $C_3^3:D_6$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(9\)\(\medspace = 3^{2} \) |
| $W$ | $C_3\times \He_3$, of order \(81\)\(\medspace = 3^{4} \) |
Related subgroups
| Centralizer: | $C_9$ | ||
| Normalizer: | $C_3^2.C_3^4$ | ||
| Complements: | $C_3^2$ $C_3^2$ | ||
| Minimal over-subgroups: | $C_3^2.C_3^3$ | $C_9.\He_3$ | $C_9.\He_3$ |
| Maximal under-subgroups: | $\He_3$ | $C_9:C_3$ | $C_3\times C_9$ |
Other information
| Number of subgroups in this autjugacy class | $27$ |
| Number of conjugacy classes in this autjugacy class | $27$ |
| Möbius function | $3$ |
| Projective image | $C_3^2\times \He_3$ |