Subgroup ($H$) information
| Description: | $C_{20}\times D_{35}$ |
| Order: | \(1400\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 7 \) |
| Index: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Exponent: | \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \) |
| Generators: |
$a^{35}b^{161}, b^{40}, a^{42}b^{70}, b^{140}, b^{70}, b^{168}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $C_{280}:C_{70}$ |
| 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: | $C_{14}$ |
| Order: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Automorphism Group: | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7$), hyperelementary, metacyclic, metabelian, a Z-group, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4\times C_{12}\times F_5\times F_7$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_4\times F_5\times F_7$ |
| $W$ | $D_{70}$, of order \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \) |
Related subgroups
Other information
| Möbius function | $1$ |
| Projective image | $C_7\times D_{70}$ |