Subgroup ($H$) information
| Description: | $C_{74}$ |
| Order: | \(74\)\(\medspace = 2 \cdot 37 \) |
| Index: | \(37\) |
| Exponent: | \(74\)\(\medspace = 2 \cdot 37 \) |
| Generators: |
$b^{37}, ab^{46}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, maximal, a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,37$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and central.
Ambient group ($G$) information
| Description: | $C_{37}\times C_{74}$ |
| Order: | \(2738\)\(\medspace = 2 \cdot 37^{2} \) |
| Exponent: | \(74\)\(\medspace = 2 \cdot 37 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 37$ (hence hyperelementary), and metacyclic.
Quotient group ($Q$) structure
| Description: | $C_{37}$ |
| Order: | \(37\) |
| Exponent: | \(37\) |
| Automorphism Group: | $C_{36}$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Outer Automorphisms: | $C_{36}$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\GL(2,37)$, of order \(1822176\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 19 \cdot 37 \) |
| $\operatorname{Aut}(H)$ | $C_{36}$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| $\operatorname{res}(S)$ | $C_{36}$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1332\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 37 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $C_{37}$ |