Subgroup ($H$) information
| Description: | $C_2\times D_{10}\times S_5$ |
| Order: | \(4800\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{2} \) |
| Index: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(2,5)(3,4), (6,8,10,7,9), (2,3)(4,5), (2,3)(4,5)(7,10)(8,9), (1,12)(2,5)(3,4)(6,8)(9,10)(11,13,14), (2,5)(3,4)(7,10)(8,9)(12,13)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian and nonsolvable.
Ambient group ($G$) information
| Description: | $A_4\times S_5^2$ |
| Order: | \(172800\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_4\times S_5\wr C_2$, of order \(691200\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $F_5.S_5\times C_2^2:S_4$ |
| $W$ | $C_3\times F_5\times S_5$, of order \(7200\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $12$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $A_4\times S_5^2$ |