Subgroup ($H$) information
| Description: | $C_{19}^2:D_9$ |
| Order: | \(6498\)\(\medspace = 2 \cdot 3^{2} \cdot 19^{2} \) |
| Index: | \(868680\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \cdot 127 \) |
| Exponent: | \(342\)\(\medspace = 2 \cdot 3^{2} \cdot 19 \) |
| Generators: |
$\left[ \left(\begin{array}{rrr}
4 & 4 & 2 \\
6 & 18 & 15 \\
12 & 3 & 18
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
1 & 14 & 17 \\
13 & 15 & 5 \\
12 & 12 & 7
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
9 & 9 & 14 \\
0 & 11 & 0 \\
16 & 4 & 13
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
5 & 17 & 10 \\
5 & 15 & 8 \\
14 & 2 & 7
\end{array}\right) \right], \left[ \left(\begin{array}{rrr}
2 & 11 & 5 \\
13 & 18 & 3 \\
14 & 4 & 18
\end{array}\right) \right]$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and an A-group.
Ambient group ($G$) information
| Description: | $\PSL(3,19)$ |
| Order: | \(5644682640\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 5 \cdot 19^{3} \cdot 127 \) |
| Exponent: | \(868680\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \cdot 127 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
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 \(33868095840\)\(\medspace = 2^{5} \cdot 3^{5} \cdot 5 \cdot 19^{3} \cdot 127 \) |
| $\operatorname{Aut}(H)$ | $C_{19}^2:(C_9\times D_{18})$, of order \(116964\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 19^{2} \) |
| $\card{W}$ | 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 | $144780$ |
| Möbius function | not computed |
| Projective image | not computed |