Subgroup ($H$) information
| Description: | $C_2^2\times D_{50}$ |
| Order: | \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Generators: |
$ad^{50}, d^{40}, c, d^{100}, b, d^{48}$
|
| Derived length: | $2$ |
The subgroup is normal, nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $\OD_{16}:D_{50}$ |
| Order: | \(1600\)\(\medspace = 2^{6} \cdot 5^{2} \) |
| Exponent: | \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_{50}\times D_4).C_{20}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $D_{25}.C_{10}\times C_2^3.\PSL(2,7)$ |
| $\card{\operatorname{res}(S)}$ | \(8000\)\(\medspace = 2^{6} \cdot 5^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_4\times D_{25}$, of order \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $D_{50}.D_4$ |