Subgroup ($H$) information
| Description: | $D_{18}$ |
| Order: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Index: | \(33\)\(\medspace = 3 \cdot 11 \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$a, b^{3}, c^{88}, c^{66}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $S_3\times D_{99}$ |
| Order: | \(1188\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 11 \) |
| Exponent: | \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3\times C_{99}).C_{30}.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_{18}:C_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
| $\operatorname{res}(S)$ | $C_9:C_6$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $W$ | $D_9$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_2$ | |||
| Normalizer: | $D_{18}$ | |||
| Normal closure: | $S_3\times D_{99}$ | |||
| Core: | $C_9$ | |||
| Minimal over-subgroups: | $D_{198}$ | $S_3\times D_9$ | ||
| Maximal under-subgroups: | $C_{18}$ | $D_9$ | $D_9$ | $D_6$ |
Other information
| Number of subgroups in this conjugacy class | $33$ |
| Möbius function | $1$ |
| Projective image | $S_3\times D_{99}$ |