Subgroup ($H$) information
| Description: | $D_5\times C_{20}$ |
| Order: | \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
| Index: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$\left(\begin{array}{rr}
0 & 274 \\
40 & 0
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 86
\end{array}\right), \left(\begin{array}{rr}
86 & 0 \\
0 & 232
\end{array}\right), \left(\begin{array}{rr}
0 & 1 \\
128 & 0
\end{array}\right), \left(\begin{array}{rr}
128 & 0 \\
0 & 191
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $D_{140}:C_{10}$ |
| Order: | \(2800\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 7 \) |
| Exponent: | \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{70}.(C_2\times C_6).C_2^6$ |
| $\operatorname{Aut}(H)$ | $D_{10}:C_4^2$, of order \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| $\operatorname{res}(S)$ | $D_{10}:C_4^2$, of order \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $W$ | $D_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $7$ |
| Möbius function | $1$ |
| Projective image | $D_{70}$ |