Subgroup ($H$) information
| Description: | $C_9\times D_9$ |
| Order: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$a, d^{3}, c^{7}, c^{3}, d$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $C_9^2:D_6$ |
| Order: | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_9^2:(C_6\times S_3)$, of order \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $C_9:C_6^2$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| $\operatorname{res}(S)$ | $C_{18}:C_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(9\)\(\medspace = 3^{2} \) |
| $W$ | $D_{18}$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_9$ | ||
| Normalizer: | $D_9^2$ | ||
| Normal closure: | $C_9^2:S_3$ | ||
| Core: | $C_9^2$ | ||
| Minimal over-subgroups: | $C_9^2:S_3$ | $D_9^2$ | |
| Maximal under-subgroups: | $C_9^2$ | $S_3\times C_9$ | $C_3\times D_9$ |
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $1$ |
| Projective image | $C_9^2:D_6$ |