Subgroup ($H$) information
| Description: | $A_4^2:C_{10}$ |
| Order: | \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| Index: | \(15\)\(\medspace = 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(5,7,8,6,9), (1,4,2), (2,3,4)(11,15)(13,14), (1,3)(2,4), (1,3)(2,4)(5,9,6,8,7) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4\times A_6.C_2^2\times S_4$ |
| $\operatorname{Aut}(H)$ | $C_4\times S_4^2$, of order \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| $W$ | $A_4\times S_4$, of order \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $15$ |
| Möbius function | $-1$ |
| Projective image | $A_4\times A_6$ |