Subgroup ($H$) information
| Description: | $\PSL(2,11)$ |
| Order: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Generators: |
$\langle(1,4)(2,8)(3,7)(5,6), (2,9,3)(4,7,6)(5,10,11)\rangle$
|
| Derived length: | $0$ |
The subgroup is characteristic (hence normal), a direct factor, nonabelian, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and an A-group.
Ambient group ($G$) information
| Description: | $S_3\times \PSL(2,11)$ |
| Order: | \(3960\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, an A-group, and nonsolvable.
Quotient group ($Q$) structure
| Description: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times \PGL(2,11)$, of order \(7920\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $\PGL(2,11)$, of order \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $\PGL(2,11)$, of order \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $\PSL(2,11)$, of order \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
Related subgroups
Other information
| Möbius function | $3$ |
| Projective image | $S_3\times \PSL(2,11)$ |