Subgroup ($H$) information
| Description: | $C_{15}\times A_6$ |
| Order: | \(5400\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5^{2} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(5,8,9,7,6)(10,11)(12,13), (1,4,2)(5,6,7,9,8)(10,15,12,11)(13,14), (1,4,2), (5,7,8,6,9)\rangle$
|
| Derived length: | $1$ |
The subgroup is maximal, nonabelian, and nonsolvable.
Ambient group ($G$) information
| Description: | $C_5\times A_4\times A_6$ |
| Order: | \(21600\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4\times A_6.C_2^2\times S_4$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_4\times A_6.C_2^2$ |
| $W$ | $A_6$, of order \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $-1$ |
| Projective image | $A_4\times A_6$ |