Subgroup ($H$) information
| Description: | $C_{15}\times D_{15}$ |
| Order: | \(450\)\(\medspace = 2 \cdot 3^{2} \cdot 5^{2} \) |
| Index: | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(2,9,3,8,7), (11,16,13)(12,14,15), (11,16,13), (1,7)(2,4)(3,6)(5,8)(9,10)(11,15)(12,13)(14,16), (1,10,5,4,6)(11,13,16)(12,15,14)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $A_5^2:S_3^2$ |
| Order: | \(129600\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\wr C_2.A_5^2.C_2^2$ |
| $\operatorname{Aut}(H)$ | $D_{30}:C_4^2$, of order \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| $W$ | $S_3\times D_{10}$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $144$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $A_5^2:S_3^2$ |