Subgroup ($H$) information
| Description: | $\PSL(2,11)$ |
| Order: | \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| Index: | \(266\)\(\medspace = 2 \cdot 7 \cdot 19 \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Generators: |
$\left(\begin{array}{rrrrrrr}
5 & 8 & 0 & 7 & 5 & 0 & 6 \\
8 & 5 & 7 & 2 & 7 & 7 & 6 \\
0 & 7 & 1 & 3 & 4 & 3 & 4 \\
7 & 2 & 3 & 4 & 2 & 5 & 9 \\
5 & 7 & 4 & 2 & 8 & 7 & 5 \\
0 & 7 & 3 & 5 & 7 & 0 & 1 \\
6 & 6 & 4 & 9 & 5 & 1 & 9
\end{array}\right), \left(\begin{array}{rrrrrrr}
9 & 10 & 8 & 7 & 6 & 6 & 3 \\
8 & 4 & 8 & 5 & 1 & 9 & 5 \\
0 & 8 & 5 & 6 & 2 & 1 & 6 \\
9 & 7 & 4 & 3 & 6 & 6 & 7 \\
1 & 1 & 2 & 9 & 3 & 0 & 9 \\
10 & 3 & 9 & 1 & 4 & 3 & 4 \\
9 & 9 & 0 & 3 & 8 & 6 & 7
\end{array}\right)$
|
| Derived length: | $0$ |
The subgroup is maximal, nonabelian, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and an A-group.
Ambient group ($G$) information
| Description: | $J_1$ |
| Order: | \(175560\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \) |
| Exponent: | \(43890\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \) |
| Derived length: | $0$ |
The ambient group is nonabelian, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and an A-group.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $J_1$, of order \(175560\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 \) |
| $\operatorname{Aut}(H)$ | $\PGL(2,11)$, of order \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| $W$ | $\PSL(2,11)$, of order \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
Related subgroups
| Centralizer: | $C_1$ | |||
| Normalizer: | $\PSL(2,11)$ | |||
| Normal closure: | $J_1$ | |||
| Core: | $C_1$ | |||
| Minimal over-subgroups: | $J_1$ | |||
| Maximal under-subgroups: | $A_5$ | $A_5$ | $C_{11}:C_5$ | $D_6$ |
Other information
| Number of subgroups in this conjugacy class | $266$ |
| Möbius function | $-1$ |
| Projective image | $J_1$ |