Subgroup ($H$) information
| Description: | $C_5:D_{20}$ |
| Order: | \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
| Index: | \(7\) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$a, c^{70}, b, c^{105}, c^{84}$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, a Hall subgroup, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_5:D_{140}$ |
| Order: | \(1400\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 7 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_5\times C_{70}).C_6.C_2^4.S_5$ |
| $\operatorname{Aut}(H)$ | $C_{10}^2:(C_2\times \GL(2,5))$, of order \(96000\)\(\medspace = 2^{8} \cdot 3 \cdot 5^{3} \) |
| $\operatorname{res}(S)$ | $C_{10}^2:(C_2\times \GL(2,5))$, of order \(96000\)\(\medspace = 2^{8} \cdot 3 \cdot 5^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_5:D_{10}$, of order \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
Related subgroups
| Centralizer: | $C_2$ | ||
| Normalizer: | $C_5:D_{20}$ | ||
| Normal closure: | $C_5:D_{140}$ | ||
| Core: | $C_5\times C_{20}$ | ||
| Minimal over-subgroups: | $C_5:D_{140}$ | ||
| Maximal under-subgroups: | $C_5\times C_{20}$ | $C_5:D_{10}$ | $D_{20}$ |
Other information
| Number of subgroups in this autjugacy class | $7$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $C_5:D_{70}$ |