Subgroup ($H$) information
| Description: | $C_{14}$ |
| Order: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Index: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$b^{20}c^{7}, c^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a semidirect factor, and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $C_{70}:Q_{16}$ |
| 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: | $C_5:Q_{16}$ |
| Order: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Automorphism Group: | $D_8:C_2\times F_5$, of order \(640\)\(\medspace = 2^{7} \cdot 5 \) |
| Outer Automorphisms: | $C_2^2\times C_4$, of order \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_{70}).C_{12}.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{W}$ | \(2\) |
Related subgroups
Other information
| Möbius function | not computed |
| Projective image | not computed |