Let $P1 = SL_{2} (\mathbb{F}_{2} [[t]])$ and $P2= \{ [a \; t^{-1}b ; tc \; d] | a,b,c,d \in \mathbb{F}_{2}[[t]], ad-bc = 1 \}$. I am considering cosets of $P1$ and $P2$. For instance, $[0 -1/t; t 0]P1$ and $[0 \; -1 ; 1\; 0 ]P2$.
I want to multiply these cosets on the left by elements of a group with generating set $\{ [1 \; 1 ; 0 \; 1], [1 \; t ; 0 \; 1] , [1 \; 0 ; 1 \; 1] [1 \; 0 ; t \; 1] , [t \; 0 ; 0 \; t^{-1}] \}$.
How can I use GAP to determine when these cosets are equal to each other?