Subgroup ($H$) information
| Description: | $\PSL(2,13)$ |
| Order: | \(1092\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 13 \) |
| Index: | \(1092\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 13 \) |
| Exponent: | \(546\)\(\medspace = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| Generators: |
$\left[ \left(\begin{array}{rrrr}
12 & 0 & 3 & 0 \\
0 & 12 & 0 & 10 \\
8 & 0 & 1 & 0 \\
0 & 5 & 0 & 1
\end{array}\right) \right], \left[ \left(\begin{array}{rrrr}
3 & 0 & 12 & 0 \\
0 & 3 & 0 & 1 \\
3 & 0 & 8 & 0 \\
0 & 10 & 0 & 8
\end{array}\right) \right]$
|
| Derived length: | $0$ |
The subgroup is normal, a direct factor, nonabelian, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and an A-group.
Ambient group ($G$) information
| Description: | $\PSL(2,13)^2$ |
| Order: | \(1192464\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2} \) |
| Exponent: | \(546\)\(\medspace = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| Derived length: | $0$ |
The ambient group is nonabelian, an A-group, and perfect (hence nonsolvable).
Quotient group ($Q$) structure
| Description: | $\PSL(2,13)$ |
| Order: | \(1092\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 13 \) |
| Exponent: | \(546\)\(\medspace = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| Automorphism Group: | $\PGL(2,13)$, of order \(2184\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Derived length: | $0$ |
The quotient 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\PGL(2,13)\wr C_2$, of order \(9539712\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2} \) |
| $\operatorname{Aut}(H)$ | $\PGL(2,13)$, of order \(2184\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \cdot 13 \) |
| $W$ | $\PSL(2,13)$, of order \(1092\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 13 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $-1092$ |
| Projective image | $\PSL(2,13)^2$ |