Subgroup ($H$) information
| Description: | $D_5^2.D_4$ |
| Order: | \(800\)\(\medspace = 2^{5} \cdot 5^{2} \) |
| Index: | \(2\) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$ae^{5}, d^{2}e^{8}, d^{5}e^{5}, b^{2}, be^{5}, e^{2}, c$
|
| Derived length: | $3$ |
The subgroup is normal, maximal, a semidirect factor, nonabelian, and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $D_{10}^2.C_2^2$ |
| Order: | \(1600\)\(\medspace = 2^{6} \cdot 5^{2} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(12800\)\(\medspace = 2^{9} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $D_{10}^2.C_2^3$, of order \(3200\)\(\medspace = 2^{7} \cdot 5^{2} \) |
| $\operatorname{res}(S)$ | $D_{10}^2.C_2^3$, of order \(3200\)\(\medspace = 2^{7} \cdot 5^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $D_5^2.C_2^3$, of order \(800\)\(\medspace = 2^{5} \cdot 5^{2} \) |
Related subgroups
Other information
| Möbius function | not computed |
| Projective image | $D_5^2.C_2^3$ |