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Using GAP interface:

G=SymmetricGroup(3); a=G((1,2)); c=G((2,3))
b, d = c, a
C = (G(x) for x in gap.AsList( gap.RightCoset(gap.Centralizer(G,c), gap.RepresentativeAction(G,c,a)) ))
try:
    x = next(x for x in C if x*b==d*x)
    print('x=',x)
except StopIteration:
    print('No x exists!')

Using GAP interface:interface and the idea from here:

G=SymmetricGroup(3); a=G((1,2)); c=G((2,3))
b, d = c, a
C = (G(x) for x in gap.AsList( gap.RightCoset(gap.Centralizer(G,c), gap.RepresentativeAction(G,c,a)) ))
try:
    x = next(x for x in C if x*b==d*x)
    print('x=',x)
except StopIteration:
    print('No x exists!')

Using GAP interface and the idea from here:

G=SymmetricGroup(3); a=G((1,2)); c=G((2,3))
b, d = c, a
a    # just as an example

C = (G(x) for x in gap.AsList( gap.RightCoset(gap.Centralizer(G,c), gap.RepresentativeAction(G,c,a)) ))
try:
    x = next(x for x in C if x*b==d*x)
    print('x=',x)
except StopIteration:
    print('No x exists!')