Subgroup ($H$) information
| Description: | $C_3^2:D_9$ |
| Order: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$a^{3}d^{2}, e^{3}, bd^{2}, de^{4}, d$
|
| Derived length: | $3$ |
The subgroup is nonabelian and supersolvable (hence solvable and monomial).
Ambient group ($G$) information
| Description: | $C_3^4.(C_3\times S_3)$ |
| Order: | \(1458\)\(\medspace = 2 \cdot 3^{6} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_3^5.C_2^2$, of order \(78732\)\(\medspace = 2^{2} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $C_3^4.S_3^2$, of order \(2916\)\(\medspace = 2^{2} \cdot 3^{6} \) |
| $\operatorname{res}(S)$ | $C_3^3.S_3^2$, of order \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(9\)\(\medspace = 3^{2} \) |
| $W$ | $\He_3.S_3$, of order \(162\)\(\medspace = 2 \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $1$ |
| Projective image | $(C_3^2\times C_9):C_6$ |