Subgroup ($H$) information
| Description: | $D_5^2$ |
| Order: | \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| Index: | \(2\) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$a^{2}, b^{2}, c, b^{5}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $D_5:F_5$ |
| Order: | \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
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
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_5\wr C_2$, of order \(800\)\(\medspace = 2^{5} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $F_5\wr C_2$, of order \(800\)\(\medspace = 2^{5} \cdot 5^{2} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $F_5\wr C_2$, of order \(800\)\(\medspace = 2^{5} \cdot 5^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $D_5:F_5$, of order \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
Related subgroups
| Centralizer: | $C_1$ | ||||
| Normalizer: | $D_5:F_5$ | ||||
| Minimal over-subgroups: | $D_5:F_5$ | ||||
| Maximal under-subgroups: | $C_5\times D_5$ | $C_5\times D_5$ | $C_5:D_5$ | $D_{10}$ | $D_{10}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $D_5:F_5$ |