Subgroup ($H$) information
| Description: | $C_5\times D_{37}$ |
| Order: | \(370\)\(\medspace = 2 \cdot 5 \cdot 37 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(370\)\(\medspace = 2 \cdot 5 \cdot 37 \) |
| Generators: |
$a, b^{444}, b^{20}$
|
| 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: | $C_5\times D_{148}$ |
| Order: | \(1480\)\(\medspace = 2^{3} \cdot 5 \cdot 37 \) |
| Exponent: | \(740\)\(\medspace = 2^{2} \cdot 5 \cdot 37 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{74}.C_{18}.C_2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_4\times F_{37}$, of order \(5328\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 37 \) |
| $\operatorname{res}(S)$ | $C_4\times F_{37}$, of order \(5328\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 37 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $D_{37}$, of order \(74\)\(\medspace = 2 \cdot 37 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $D_{148}$ |