Subgroup ($H$) information
| Description: | $C_2\times D_6\times F_5$ |
| Order: | \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Index: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(6,9,10,8,7)(11,14), (2,3)(4,5), (1,12)(2,3)(4,5)(6,10,9,7), (7,9)(8,10), (6,7,9,10)(11,14,13), (2,3)(4,5)(6,9,10,8,7), (2,3)(4,5)(7,10,9,8)\rangle$
|
| 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: | $A_4\times S_5^2$ |
| 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 \(691200\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $C_2^4:S_3^2\times F_5$, of order \(11520\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \) |
| $W$ | $S_3\times F_5$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $360$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $A_4\times S_5^2$ |