ExampleMagmaCode8.mgm

// Want to find the "distinct" necklaces
// with 8 black/white beads.
// Two necklaces are the "same"
// if an element of D8 changes one to the other.
// Represent a necklace by the set of black beads.
D8 := DihedralGroup(8);

// Considerr finding the necklaces with two black beads.
N2 := { {a,b} : a,b in [1..8] | a ne b};
#N2;
N2;

t := Rep(N2);
t;
O := Orbit(D8, t);
O;

t2 := rep{ a : a in N2 | a notin O};
t2;
O2 := Orbit(D8, t2);
O2;

t3 := rep{ a : a in N2 | a notin O and a notin O2};
t3;
O3 := Orbit(D8, t3);
O3;

t4 := rep{ a : a in N2 | a notin O and a notin O2 and a notin O3};
t4;
O4 := Orbit(D8, t4);
O4;

#N2;
#O + #O2 + #O3 + #O4;
t; t2; t3; t4;

// Now let's consider turning this into a loop
// that does the work for us.
N2 := { {a,b} : a,b in [1..8] | a ne b};
DistinctNecklaces := {};
IsEmpty(N2);
t := Rep(N2);
Include(~DistinctNecklaces, t);
O := Orbit(D8,t);X
DistinctNecklaces;
for o in O do
  Exclude(~N2,o);
end for;
N2;
#N2;
O;
IsEmpty(N2);
t := Rep(N2);
Include(~DistinctNecklaces, t);
O := Orbit(D8,t);X
for o in O do
  Exclude(~N2,o);
end for;
for o in O do
  Exclude(~N2,o);
end for;
DistinctNecklaces;
#N2;

// All together now:
N2 := { {a,b} : a,b in [1..8] | a ne b};
DistinctNecklaces := {};
while not IsEmpty(N2) do
  t := Rep(N2);
  Include(~DistinctNecklaces, t);
  O := Orbit(D8,t);
  for o in O do
    Exclude(~N2,o);
  end for;
end while;
N2;
DistinctNecklaces;

// Can do the same for necklaces with 3 black beads:
N3 := { {a,b,c} : a,b,c in [1..8] | #{a,b,c} eq 3};
DistinctNecklaces := {};
#N3;
while not IsEmpty(N3) do
  t := Rep(N3);
  Include(~DistinctNecklaces, t);
  O := Orbit(D8,t);
  for o in O do
    Exclude(~N3,o);
  end for;
end while;
N3;
DistinctNecklaces;