Subgroup ($H$) information
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(370\)\(\medspace = 2 \cdot 5 \cdot 37 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$a^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the Frattini subgroup (hence characteristic and normal), cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, and a $p$-group.
Ambient group ($G$) information
| Description: | $C_{37}:C_{40}$ |
| Order: | \(1480\)\(\medspace = 2^{3} \cdot 5 \cdot 37 \) |
| Exponent: | \(1480\)\(\medspace = 2^{3} \cdot 5 \cdot 37 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Quotient group ($Q$) structure
| Description: | $C_5\times D_{37}$ |
| Order: | \(370\)\(\medspace = 2 \cdot 5 \cdot 37 \) |
| Exponent: | \(370\)\(\medspace = 2 \cdot 5 \cdot 37 \) |
| Automorphism Group: | $C_4\times F_{37}$, of order \(5328\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 37 \) |
| Outer Automorphisms: | $C_2\times C_{36}$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{37}.C_{18}.C_2.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(10656\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 37 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{37}:C_{40}$ | ||
| Normalizer: | $C_{37}:C_{40}$ | ||
| Minimal over-subgroups: | $C_{148}$ | $C_{20}$ | $C_8$ |
| Maximal under-subgroups: | $C_2$ |
Other information
| Möbius function | $-37$ |
| Projective image | $C_5\times D_{37}$ |