Subgroup ($H$) information
| Description: | $C_2\times S_4\times A_5$ |
| Order: | \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| Index: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(7,9), (1,9,5,6,7,8)(2,3,4)(10,11)(12,13), (1,6)(7,9), (5,8)(7,9), (1,8,7,6,5,9)(2,4,3)(10,14,13), (1,8)(3,4)(5,6)(7,9), (1,8,9)(2,4,3)(5,7,6)\rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and nonsolvable.
Ambient group ($G$) information
| Description: | $A_5\times S_3\wr S_3$ |
| Order: | \(77760\)\(\medspace = 2^{6} \cdot 3^{5} \cdot 5 \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\wr S_3\times S_5$, of order \(155520\)\(\medspace = 2^{7} \cdot 3^{5} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_2\times S_4\times S_5$, of order \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) |
| $W$ | $S_4\times A_5$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $27$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $A_5\times S_3\wr S_3$ |