Subgroup ($H$) information
| Description: | $C_9^2$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Generators: |
$\left(\begin{array}{rr}
19 & 63 \\
27 & 10
\end{array}\right), \left(\begin{array}{rr}
28 & 18 \\
54 & 55
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_9^2.C_3^2$ |
| Order: | \(729\)\(\medspace = 3^{6} \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Nilpotency class: | $4$ |
| 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_9^2.C_3^4.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_3^4:\GL(2,3)$, of order \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_3^2:D_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(243\)\(\medspace = 3^{5} \) |
| $W$ | $C_3$, of order \(3\) |
Related subgroups
| Centralizer: | $C_3\times C_9^2$ | |||
| Normalizer: | $C_9^2.C_3^2$ | |||
| Minimal over-subgroups: | $C_3\times C_9^2$ | $C_9^2:C_3$ | $C_9^2.C_3$ | $C_9^2.C_3$ |
| Maximal under-subgroups: | $C_3\times C_9$ | $C_3\times C_9$ |
Other information
| Möbius function | $3$ |
| Projective image | $\He_3:C_3^2$ |