Subgroup ($H$) information
| Description: | $A_4\times A_5$ |
| Order: | \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
| Index: | \(2\) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(6,9,8), (6,9)(7,8), (1,3,5)(6,7,9), (1,2)(3,4)(6,8)(7,9), (6,8)(7,9)\rangle$
|
| Derived length: | $2$ |
The subgroup is the commutator subgroup (hence characteristic and normal), maximal, a semidirect factor, nonabelian, an A-group, and nonsolvable.
Ambient group ($G$) information
| Description: | $S_4\times A_5$ |
| Order: | \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_2$ |
| Order: | \(2\) |
| Exponent: | \(2\) |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, simple, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_4\times S_5$, of order \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $S_4\times S_5$, of order \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_4\times S_5$, of order \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $S_4\times A_5$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_1$ | ||||
| Normalizer: | $S_4\times A_5$ | ||||
| Complements: | $C_2$ $C_2$ | ||||
| Minimal over-subgroups: | $S_4\times A_5$ | ||||
| Maximal under-subgroups: | $C_2^2\times A_5$ | $\GL(2,4)$ | $A_4^2$ | $D_5\times A_4$ | $S_3\times A_4$ |
Other information
| Möbius function | $-1$ |
| Projective image | $S_4\times A_5$ |