Let V a vector space of dimension 2 over $\Bbb F^3$, i.e $V = \Bbb F^2_3$. We know that $SL_2(\Bbb F^3)$ acts on $\Omega = \{W \le V : \dim(W) = 1\}$. Note that $card(\Omega)= 4$.
It implies that there exists an homomorphism $$\theta : SL_2(\Bbb F^3) \longrightarrow \operatorname{Sym}(\Omega) \simeq S_4$$
$$g \longmapsto \theta (g)(x) = gx$$
Note that $\ker(\theta) = Z = \{kI_2$ with k $\in \Bbb F^3 \}$. Right now I am trying to understand the solution given to an exercice. They numbered the four elements of $\Omega$ as follow:
1 = $<(1,0)>$
2 = $<(0,1)>$
3 = $<(1,1)>$
4 = $<(1,2)>$
Then, they wrote that we can compute some images by $\theta$ :
$$\theta(\begin{pmatrix} 1 & 1\\ 0 & 1\\ \end{pmatrix})=(234)$$ $$\theta(\begin{pmatrix} 1 & 0\\ 1 & 1\\ \end{pmatrix})=(134)$$
They did not provide any steps for the computation, and I do not understand how $SL_2(\Bbb F^3)$ acts on $\Omega$. Furthermore in the next question they ask to compute the inverse of some groups $H$, $H = <(123)>, H = <(234)>, H = <(12)(34)>$:
$$\theta^{-1}((12)(34))=\begin{pmatrix} 1 & 1\\ 0 & 1\\ \end{pmatrix}Z$$