Subgroup ($H$) information
| Description: | $C_3^2\times D_9$ |
| Order: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Index: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$d^{3}, f^{3}, cd^{2}, d^{2}, e^{3}f$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_3^5.S_3^2:C_2^2$ |
| Order: | \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_9^2.(C_3\times C_6).C_4.C_6.C_2^3$, of order \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $C_9:C_6\times \GL(2,3)$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| $W$ | $C_{18}:C_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $72$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_3^5.S_3^2:C_2^2$ |