Subgroup ($H$) information
| Description: | $\PSL(2,13)$ |
| Order: | \(1092\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 13 \) |
| Index: | \(2184\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13 \) |
| Exponent: | \(546\)\(\medspace = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| Generators: |
$\left(\begin{array}{rrrr}
2 & 9 & 0 & 0 \\
4 & 11 & 0 & 0 \\
7 & 12 & 11 & 9 \\
12 & 7 & 4 & 2
\end{array}\right), \left(\begin{array}{rrrr}
8 & 1 & 9 & 7 \\
3 & 9 & 5 & 11 \\
4 & 7 & 0 & 0 \\
5 & 2 & 0 & 0
\end{array}\right)$
|
| Derived length: | $0$ |
The subgroup is nonabelian, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and an A-group.
Ambient group ($G$) information
| Description: | $\OmegaPlus(4,13)$ |
| Order: | \(2384928\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2} \) |
| Exponent: | \(1092\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 13 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and perfect (hence nonsolvable). Whether it is almost simple has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PGL(2,13)\wr C_2$, of order \(9539712\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2} \) |
| $\operatorname{Aut}(H)$ | $\PGL(2,13)$, of order \(2184\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $1092$ |
| Möbius function | not computed |
| Projective image | not computed |