Subgroup ($H$) information
| Description: | $C_2^3\times D_{10}$ |
| Order: | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Index: | \(270\)\(\medspace = 2 \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$\langle(2,5)(3,4), (2,3)(4,5), (2,5)(3,4)(11,13)(12,14), (2,3)(4,5)(11,14)(12,13), (6,7,10,9,8)(11,14)(12,13), (2,3)(4,5)(6,9)(7,10)(11,12)(13,14)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $A_4\times A_5^2$ |
| Order: | \(43200\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, an A-group, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_4\times S_5\wr C_2$, of order \(691200\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $C_2^4.A_8\times F_5$, of order \(6451200\)\(\medspace = 2^{12} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
| $W$ | $C_3^2\times D_5$, of order \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $60$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $A_4\times A_5^2$ |