Subgroup ($H$) information
| Description: | $C_2\times A_4\times S_5$ |
| Order: | \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(7,8,9)(10,13)(11,12), (6,8)(7,9), (6,9)(7,8), (1,2)(6,9,8)(10,11)(12,13), (1,5,4)(2,3)(6,9)(7,8)(10,11)(12,13), (6,9,8)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian and nonsolvable.
Ambient group ($G$) information
| Description: | $D_4\times A_4\times S_5$ |
| Order: | \(11520\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$, of order \(46080\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2\times S_4\times S_5$, of order \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) |
| $W$ | $A_4\times S_5$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $8$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | not computed |
| Projective image | $D_4\times A_4\times S_5$ |