Subgroup ($H$) information
| Description: | $C_2\times F_{19}$ |
| Order: | \(684\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 19 \) |
| Index: | \(38\)\(\medspace = 2 \cdot 19 \) |
| Exponent: | \(342\)\(\medspace = 2 \cdot 3^{2} \cdot 19 \) |
| Generators: |
$a^{6}, d, a^{2}, b^{2}c^{14}d^{7}, a^{9}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
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).
Quotient set structure
Since this subgroup has trivial core, the ambient group $G$ acts faithfully and transitively on the set of cosets of $H$. The resulting permutation representation is isomorphic to 38T36.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{19}^2.C_{36}.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_2\times F_{19}$, of order \(684\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 19 \) |
| $W$ | $F_{19}$, 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 | $D_{19}^2:C_{18}$ |