Subgroup ($H$) information
| Description: | $C_{20}\times D_{35}$ |
| Order: | \(1400\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 7 \) |
| Index: | \(2\) |
| Exponent: | \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \) |
| Generators: |
$\left(\begin{array}{rr}
79 & 0 \\
0 & 249
\end{array}\right), \left(\begin{array}{rr}
0 & 9 \\
125 & 0
\end{array}\right), \left(\begin{array}{rr}
1 & 0 \\
0 & 86
\end{array}\right), \left(\begin{array}{rr}
86 & 0 \\
0 & 232
\end{array}\right), \left(\begin{array}{rr}
0 & 1 \\
128 & 0
\end{array}\right), \left(\begin{array}{rr}
128 & 0 \\
0 & 191
\end{array}\right)$
|
| Derived length: | $2$ |
The subgroup is normal, maximal, a semidirect factor, 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.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{70}.(C_2\times C_6).C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times C_4\times F_5\times F_7$ |
| $\card{\operatorname{res}(S)}$ | \(13440\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $D_{70}$, of order \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $D_{70}$ |