Subgroup ($H$) information
| Description: | $C_2\times C_{20}$ |
| Order: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Index: | \(980\)\(\medspace = 2^{2} \cdot 5 \cdot 7^{2} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$a^{70}b^{245}, a^{84}b^{210}, b^{140}, b^{210}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_{140}.C_{280}$ |
| Order: | \(39200\)\(\medspace = 2^{5} \cdot 5^{2} \cdot 7^{2} \) |
| Exponent: | \(560\)\(\medspace = 2^{4} \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and metacyclic (hence solvable, supersolvable, monomial, and metabelian).
Quotient group ($Q$) structure
| Description: | $C_{35}:C_{28}$ |
| Order: | \(980\)\(\medspace = 2^{2} \cdot 5 \cdot 7^{2} \) |
| Exponent: | \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \) |
| Automorphism Group: | $C_2\times C_6\times F_5\times F_7$ |
| Outer Automorphisms: | $C_2\times C_6\times C_{12}$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(645120\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_4\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Möbius function | not computed |
| Projective image | not computed |