Orbits of a matrix group

Question

How to get the orbits of the action of the following matrix group on the standard basis of a 3-dim vector space? \begin{pmatrix} SL_2(2) & 0\\ * & 1\\ \end{pmatrix}

where * denotes a 1$\times$2 matrix with arbitrary entries in the field of 2 elements.

I suppose there are two orbits, one the zero vector and the other consisting of all other vectors?

Answer 1

If $G$ is the matrix group actin on vector space $V$, you can get the orbit of $G$ on a specific vector $v\in V$ with $\mathtt{Orbit(G,v)}$.

I don't think there is a single command to return all of the orbits. An easy way to do this is

$\mathtt{\{Orbit(G,v): v\ in\ V\}}$,

although that would not be very efficient when $V$ is large.

You can construct the group as follows:

X := SL(3,2);
V := VectorSpace(X);
G := Stabilizer(X, sub<V|V.1,V.2> );