Subgroup ($H$) information
| Description: | $C_{19}:C_9$ |
| Order: | \(171\)\(\medspace = 3^{2} \cdot 19 \) |
| Index: | \(152\)\(\medspace = 2^{3} \cdot 19 \) |
| Exponent: | \(171\)\(\medspace = 3^{2} \cdot 19 \) |
| Generators: |
$a^{2}, a^{6}, d$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 3$.
Ambient group ($G$) information
| Description: | $D_{19}^2:C_{18}$ |
| Order: | \(25992\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 19^{2} \) |
| Exponent: | \(684\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 19 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{19}^2.C_{36}.C_2^2$ |
| $\operatorname{Aut}(H)$ | $F_{19}$, of order \(342\)\(\medspace = 2 \cdot 3^{2} \cdot 19 \) |
| $W$ | $F_{19}$, of order \(342\)\(\medspace = 2 \cdot 3^{2} \cdot 19 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $38$ |
| Möbius function | $0$ |
| Projective image | $D_{19}^2:C_{18}$ |