Subgroup ($H$) information
| Description: | $D_5\times A_4$ |
| Order: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Index: | \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,2)(3,4), (5,8,9,6,7), (1,3)(2,4), (1,2,4), (5,7)(6,8)\rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $A_4\times F_5\times S_6$ |
| Order: | \(172800\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_2\times S_6$ |
| Order: | \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Automorphism Group: | $S_6:C_2^2$, of order \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $1$ |
The quotient is nonabelian, nonsolvable, and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_5.A_6.C_2^2\times S_4$ |
| $\operatorname{Aut}(H)$ | $F_5\times S_4$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| $W$ | $A_4\times F_5$, of order \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
Related subgroups
| Centralizer: | $S_6$ | |||
| Normalizer: | $A_4\times F_5\times S_6$ | |||
| Minimal over-subgroups: | $C_{10}^2:C_6$ | $A_4\times F_5$ | $A_4\times F_5$ | $A_4\times F_5$ |
| Maximal under-subgroups: | $C_5\times A_4$ | $C_2\times D_{10}$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $A_4\times F_5\times S_6$ |