Subgroup ($H$) information
| Description: | $S_3\times A_5^2$ |
| Order: | \(21600\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{2} \) |
| Index: | $1$ |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(2,3), (1,5,11,12,13)(3,4)(6,9,7,10,8), (2,3,4), (2,3)(5,12)(6,9)(7,10)(11,13)\rangle$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a direct factor, nonabelian, a Hall subgroup, an A-group, and nonsolvable.
Ambient group ($G$) information
| Description: | $S_3\times A_5^2$ |
| Order: | \(21600\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, an A-group, and nonsolvable.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_5^2:D_6$, of order \(172800\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $S_5^2:D_6$, of order \(172800\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5^{2} \) |
| $W$ | $S_3\times A_5^2$, of order \(21600\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{2} \) |
Related subgroups
Other information
| Möbius function | $1$ |
| Projective image | $S_3\times A_5^2$ |