Subgroup ($H$) information
| Description: | $C_6:D_{18}$ |
| Order: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$a, e^{3}, e^{4}, c^{3}, d^{3}, d^{2}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_2\times C_3^3:D_{18}$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^2.(C_3^3\times Q_8).C_3^4.C_2^5$ |
| $\operatorname{Aut}(H)$ | $S_4\times C_3^4.S_3^2$, of order \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \) |
| $\operatorname{res}(S)$ | $C_3^4.D_6^2$, of order \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $W$ | $C_3:D_9$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $9$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $3$ |
| Projective image | $C_3^3:D_{18}$ |