Subgroup ($H$) information
| Description: | not computed |
| Order: | \(14815440\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5 \cdot 19^{3} \) |
| Index: | \(1143\)\(\medspace = 3^{2} \cdot 127 \) |
| Exponent: | not computed |
| Generators: |
$\left[ \left(\begin{array}{rrr}
4 & 2 & 11 \\
1 & 7 & 1 \\
13 & 11 & 6
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
7 & 13 & 5 \\
16 & 17 & 16 \\
11 & 2 & 13
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
17 & 2 & 8 \\
15 & 14 & 3 \\
6 & 5 & 12
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
0 & 16 & 12 \\
13 & 18 & 13 \\
15 & 1 & 3
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
15 & 0 & 15 \\
13 & 12 & 14 \\
5 & 0 & 10
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
1 & 2 & 11 \\
1 & 4 & 1 \\
14 & 6 & 4
\end{array}\right) \right]$
|
| Derived length: | not computed |
The subgroup is nonabelian and nonsolvable. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $\PGL(3,19)$ |
| Order: | \(16934047920\)\(\medspace = 2^{4} \cdot 3^{5} \cdot 5 \cdot 19^{3} \cdot 127 \) |
| Exponent: | \(868680\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \cdot 127 \) |
| Derived length: | $1$ |
The ambient group is nonabelian, almost simple, and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(33868095840\)\(\medspace = 2^{5} \cdot 3^{5} \cdot 5 \cdot 19^{3} \cdot 127 \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | 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 | $381$ |
| Möbius function | not computed |
| Projective image | not computed |