Subgroup ($H$) information
| Description: | $C_9:C_3^2$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Generators: |
$a^{2}de, bc^{2}e^{6}, ce^{6}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_2\times C_3^3.C_3^3$ |
| Order: | \(1458\)\(\medspace = 2 \cdot 3^{6} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 3$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_3\times C_6$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| 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), elementary for $p = 3$ (hence hyperelementary), and metacyclic.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_3^5.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_3^4:S_3^2$, of order \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_3^3:S_3^2$, of order \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(27\)\(\medspace = 3^{3} \) |
| $W$ | $C_3\times \He_3$, of order \(81\)\(\medspace = 3^{4} \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-3$ |
| Projective image | $C_3^4:C_6$ |