Subgroup ($H$) information
| Description: | $C_{20}\times D_{35}$ |
| Order: | \(1400\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 7 \) |
| Index: | \(112\)\(\medspace = 2^{4} \cdot 7 \) |
| Exponent: | \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rr}
9 & 0 \\
0 & 224
\end{array}\right), \left(\begin{array}{rr}
0 & 125 \\
9 & 0
\end{array}\right), \left(\begin{array}{rr}
81 & 0 \\
0 & 170
\end{array}\right), \left(\begin{array}{rr}
249 & 0 \\
0 & 79
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 153
\end{array}\right), \left(\begin{array}{rr}
98 & 0 \\
0 & 238
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $C_{280}.D_{280}$ |
| Order: | \(156800\)\(\medspace = 2^{7} \cdot 5^{2} \cdot 7^{2} \) |
| Exponent: | \(560\)\(\medspace = 2^{4} \cdot 5 \cdot 7 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(5160960\)\(\medspace = 2^{14} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_4\times F_5\times F_7$ |
| $W$ | $D_{70}$, of order \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_{280}$ |
| Normalizer: | $C_{280}.D_{70}$ |
| Normal closure: | $D_{280}:C_{10}$ |
| Core: | $C_5\times C_{140}$ |
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | not computed |