Subgroup ($H$) information
| Description: | $C_{140}$ |
| Order: | \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \) |
| Index: | \(560\)\(\medspace = 2^{4} \cdot 5 \cdot 7 \) |
| Exponent: | \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rr}
9 & 0 \\
0 & 9
\end{array}\right), \left(\begin{array}{rr}
280 & 0 \\
0 & 280
\end{array}\right), \left(\begin{array}{rr}
165 & 0 \\
0 & 165
\end{array}\right), \left(\begin{array}{rr}
200 & 0 \\
0 & 200
\end{array}\right)$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal) and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group). Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_{280}.D_{140}$ |
| Order: | \(78400\)\(\medspace = 2^{6} \cdot 5^{2} \cdot 7^{2} \) |
| Exponent: | \(280\)\(\medspace = 2^{3} \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2\times D_{140}$ |
| Order: | \(560\)\(\medspace = 2^{4} \cdot 5 \cdot 7 \) |
| Exponent: | \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \) |
| Automorphism Group: | $C_{70}.(C_2^3\times C_6).C_2^4$ |
| Outer Automorphisms: | $C_2^2\wr C_2\times C_{12}$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(2580480\)\(\medspace = 2^{13} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_{12}$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $\card{W}$ | 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 |