Subgroup ($H$) information
| Description: | $C_2\times C_{10}$ |
| Order: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Index: | \(980\)\(\medspace = 2^{2} \cdot 5 \cdot 7^{2} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$a^{70}b^{105}, a^{84}b^{70}, b^{70}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_{140}.C_{140}$ |
| Order: | \(19600\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 7^{2} \) |
| Exponent: | \(280\)\(\medspace = 2^{3} \cdot 5 \cdot 7 \) |
| 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{70}.C_6^2.C_2^6.C_2$ |
| $\operatorname{Aut}(H)$ | $C_4\times S_3$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_{70}:C_{28}$ |