Subgroup ($H$) information
| Description: | $C_7:C_{40}$ |
| Order: | \(280\)\(\medspace = 2^{3} \cdot 5 \cdot 7 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(280\)\(\medspace = 2^{3} \cdot 5 \cdot 7 \) |
| Generators: |
$ad^{95}, bd^{70}, d^{70}, d^{84}, d^{20}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $D_{28}.D_{10}$ |
| Order: | \(1120\)\(\medspace = 2^{5} \cdot 5 \cdot 7 \) |
| Exponent: | \(280\)\(\medspace = 2^{3} \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7.(C_{12}\times F_5).C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_4\times C_2^2\times F_7$, of order \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_4\times C_2^2\times F_7$, of order \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $W$ | $C_2\times D_{14}$, of order \(56\)\(\medspace = 2^{3} \cdot 7 \) |
Related subgroups
| Centralizer: | $C_{20}$ | ||
| Normalizer: | $D_{28}.D_{10}$ | ||
| Minimal over-subgroups: | $C_{35}:\OD_{16}$ | $C_{35}:\SD_{16}$ | $C_{35}:Q_{16}$ |
| Maximal under-subgroups: | $C_{140}$ | $C_7:C_8$ | $C_{40}$ |
Other information
| Möbius function | $2$ |
| Projective image | $D_{10}:D_{14}$ |