Subgroup ($H$) information
| Description: | $D_{10}:D_{10}$ |
| Order: | \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$\langle(1,4,5,3,2), (11,13)(12,14), (6,7,8,10,9), (11,12)(13,14), (7,9)(8,10)(12,14), (1,4)(2,5)(6,7)(8,9)(11,13)\rangle$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_5:D_4\times A_5$ |
| Order: | \(2400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\times F_5\times S_5$, of order \(9600\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $D_5^2.C_2^5$ |
| $\operatorname{res}(S)$ | $C_2^2\times F_5^2$, of order \(1600\)\(\medspace = 2^{6} \cdot 5^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $D_5\times D_{10}$, of order \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $6$ |
| Möbius function | $-1$ |
| Projective image | $D_{10}\times A_5$ |