Subgroup ($H$) information
| Description: | $C_5$ |
| Order: | \(5\) |
| Index: | \(224\)\(\medspace = 2^{5} \cdot 7 \) |
| Exponent: | \(5\) |
| Generators: |
$c^{168}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $5$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_4.D_{140}$ |
| 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: | $C_4.D_{28}$ |
| Order: | \(224\)\(\medspace = 2^{5} \cdot 7 \) |
| Exponent: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Automorphism Group: | $F_7\times D_4^2$, of order \(2688\)\(\medspace = 2^{7} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_2^2\times C_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_{70}).C_6.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(13440\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \cdot 7 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $\OD_{16}\times C_{35}$ | |||
| Normalizer: | $C_4.D_{140}$ | |||
| Complements: | $C_4.D_{28}$ | |||
| Minimal over-subgroups: | $C_{35}$ | $C_{10}$ | $C_{10}$ | $D_5$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_4.D_{140}$ |