Subgroup ($H$) information
| Description: | $A_6.S_5$ |
| Order: | \(43200\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{2} \) |
| Index: | \(2\) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,4)(7,8,9,13)(11,14,12,15), (1,2,3,4,5)(6,8)(7,10)(9,11)(13,15)\rangle$
|
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), maximal, a semidirect factor, nonabelian, and nonsolvable.
Ambient group ($G$) information
| Description: | $S_6:S_5$ |
| Order: | \(86400\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $1$ |
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_5\times S_6:C_2$, of order \(172800\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $S_5\times S_6:C_2$, of order \(172800\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5^{2} \) |
| $W$ | $S_6:S_5$, of order \(86400\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5^{2} \) |
Related subgroups
Other information
| Möbius function | $-1$ |
| Projective image | $S_6:S_5$ |