Subgroup ($H$) information
| Description: | $D_{18}:C_6$ |
| Order: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Index: | \(3\) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$a, d^{6}, d^{8}, c^{3}, c^{2}d^{6}, d^{9}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $(C_3^2\times A_4):S_3$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_9:C_3^2:S_4$, of order \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $D_{18}:C_6$, of order \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| $\operatorname{res}(S)$ | $C_{18}:C_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_{18}:C_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $-1$ |
| Projective image | $(C_3^2\times A_4):S_3$ |