assume not starting from resolvent of quartic,

how to find [[1,2],[3,4]] and [[1,3],[2,4]] and [[1,4],[2,3]] ?

a1*a2 + a3*a4 if [[1,3,2,4],[1,2]]

a1*a3 + a2*a4 if [2,3][[1,3,2,4],[1,2]]

a1*a4 + a2*a3 if [2,4],[[1,3,2,4],[1,2]]

moreover, i use multiplication of permutation group can not multiply below

though i know how to operate multiplication by hand, i follow the syntax to do multiplication, seems not the way

with(group):

mulperms([[2,3]], [[1,3,2,4],[1,2]]);

mulperms([[1,3,2,4],[1,2]], [[2,3]]);

mulperms([2,3], [[1,3,2,4],[1,2]]);

mulperms([[1,3,2,4],[1,2]], [2,3]);