code: def test_exponent_vector():
      Groups = [
          SymmetricGroup(3),
          SymmetricGroup(4),
          SymmetricGroup(9).sylow_subgroup(3),
          SymmetricGroup(9).sylow_subgroup(2),
          SymmetricGroup(8).sylow_subgroup(2),
      ]
  
      for G in Groups:
          PcGroup = G.polycyclic_group()
          collector = PcGroup.collector
  
          pcgs = PcGroup.pcgs
          # free_group = collector.free_group
  
          for gen in G.generators:
              exp = collector.exponent_vector(gen)
              g = Permutation()
              for i in range(len(exp)):
                  g = g * pcgs[i] ** exp[i] if exp[i] else g
              assert g == gen
  
imports: from sympy.combinatorics.permutations import Permutation
  from sympy.combinatorics.named_groups import SymmetricGroup, AlternatingGroup, DihedralGroup
  from sympy.matrices import Matrix
label_desc: Use image node_id 2 for calling a global function with example usage: test_exponent_vector() without return types