Subgroup ($H$) information
| Description: | $A_4\times C_{15}$ |
| Order: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Index: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(4,9,10), (4,8)(9,10), (1,7,2,5,3)(4,10)(8,9), (4,10)(8,9), (11,12,13)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $A_5^2:C_6$ |
| Order: | \(21600\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $1$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_5^2:C_2^3$, of order \(28800\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $C_4\times S_3\times S_4$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| $W$ | $C_2\times A_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $60$ |
| Möbius function | $0$ |
| Projective image | $\SOPlus(4,4)$ |