Subgroup ($H$) information
| Description: | $C_9^2$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Generators: |
$\left(\begin{array}{rr}
46 & 36 \\
72 & 10
\end{array}\right), \left(\begin{array}{rr}
37 & 27 \\
54 & 46
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is 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: | $C_9\wr C_3$ |
| Order: | \(2187\)\(\medspace = 3^{7} \) |
| Exponent: | \(27\)\(\medspace = 3^{3} \) |
| Nilpotency class: | $5$ |
| Derived length: | $2$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Quotient set structure
Since this subgroup has trivial core, the ambient group $G$ acts faithfully and transitively on the set of cosets of $H$. The resulting permutation representation is isomorphic to 27T434.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_9^2.C_3^3.C_6^2$ |
| $\operatorname{Aut}(H)$ | $C_3^4:\GL(2,3)$, of order \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| $\operatorname{res}(S)$ | $C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(81\)\(\medspace = 3^{4} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_9^3$ | ||
| Normalizer: | $C_9^3$ | ||
| Normal closure: | $C_9^3$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_3\times C_9^2$ | ||
| Maximal under-subgroups: | $C_3\times C_9$ | $C_3\times C_9$ | $C_3\times C_9$ |
Other information
| Number of subgroups in this autjugacy class | $81$ |
| Number of conjugacy classes in this autjugacy class | $27$ |
| Möbius function | $0$ |
| Projective image | $C_9\wr C_3$ |