Subgroup ($H$) information
| Description: | $D_{37}$ |
| Order: | \(74\)\(\medspace = 2 \cdot 37 \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(74\)\(\medspace = 2 \cdot 37 \) |
| Generators: |
$a^{2}bc^{118}, c^{4}$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{74}.C_4^2$ |
| Order: | \(1184\)\(\medspace = 2^{5} \cdot 37 \) |
| Exponent: | \(148\)\(\medspace = 2^{2} \cdot 37 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2^2:C_4$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2^2\wr C_2$, of order \(32\)\(\medspace = 2^{5} \) |
| Outer Automorphisms: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{74}.C_{18}.C_4.C_2^3$ |
| $\operatorname{Aut}(H)$ | $F_{37}$, of order \(1332\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 37 \) |
| $\operatorname{res}(S)$ | $F_{37}$, of order \(1332\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 37 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $C_{37}:C_4$, of order \(148\)\(\medspace = 2^{2} \cdot 37 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_{74}.C_4^2$ |