Subgroup ($H$) information
| Description: | $C_{19}:C_4$ |
| Order: | \(76\)\(\medspace = 2^{2} \cdot 19 \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(76\)\(\medspace = 2^{2} \cdot 19 \) |
| Generators: |
$b, b^{2}, d^{2}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $(C_2\times C_4).D_{76}$ |
| Order: | \(1216\)\(\medspace = 2^{6} \cdot 19 \) |
| Exponent: | \(152\)\(\medspace = 2^{3} \cdot 19 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{19}.(C_{18}\times D_4).C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2\times F_{19}$, of order \(684\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 19 \) |
| $\operatorname{res}(S)$ | $C_2\times F_{19}$, of order \(684\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 19 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $D_{38}$, of order \(76\)\(\medspace = 2^{2} \cdot 19 \) |
Related subgroups
| Centralizer: | $C_2^2$ | |
| Normalizer: | $C_{38}.D_4$ | |
| Normal closure: | $C_{38}.D_4$ | |
| Core: | $C_{19}$ | |
| Minimal over-subgroups: | $C_{38}:C_4$ | |
| Maximal under-subgroups: | $C_{38}$ | $C_4$ |
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $(C_2\times C_4).D_{76}$ |