Subgroup ($H$) information
| Description: | $C_4\times D_5$ |
| Order: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Index: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$b, d^{70}, d^{84}, d^{105}$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
Ambient group ($G$) information
| Description: | $C_{140}.C_2^3$ |
| Order: | \(1120\)\(\medspace = 2^{5} \cdot 5 \cdot 7 \) |
| Exponent: | \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $D_{14}$ |
| Order: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Automorphism Group: | $C_2\times F_7$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Outer Automorphisms: | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times F_5\times S_4\times F_7$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times F_5$, of order \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| $\operatorname{res}(S)$ | $C_2^2\times F_5$, of order \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| $W$ | $D_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | $-14$ |
| Projective image | $D_{10}\times D_{14}$ |