Prod := function(x);
 return &*x; end function;

/* This function checks if the product of a tuple satisfies the eigenvalue condition */
P := function(X, k: A := {1}, Q := X[1]);
error if Q notin X, "Q is not in X";
Y := Exclude(X, Q); G := Sym(SetToIndexedSet(Seqset(Y))); /* just to reduce permutations checked*/
if exists(g){g: g in G| exists(s){s : s in Eigenvalues(Q*Prod(Eltseq(g)))|s[1] notin A and 
s[2] eq k }} then;
 return true, [Q] cat Eltseq(g); /* sequence returned is the order to take the product in */ 
 else return false, _;
 end if;
 end function;

function Sub_Order(W, x); 
G := sub<W | x>;
return #G; 
end function;

function Conj_Classes(W); 
s := 0;
Q := [W.1];
repeat s := s + 1; 
if exists(q){q: q in W| IsReflection(q) and forall(w){w: w in Q | not IsConjugate(W, q, w)}} then; 
Append(~Q, q); 
s := 0;
end if;
until s eq 1;
return Q;
end function;

function Fix_M(W: T := Rank(W) + 1, Q := [W.1]);
i := 0; 
Total := {};
repeat i := i + 1; 
X := Subsets(IndexedSetToSet(&join{Class(W, q) : q in Q[i..#Q]} diff Seqset(Q[1..i])), T-1);
X_1 :=  &join{{{Q[i]} join x} : x in X}; 
Total := Total join X_1; /*X_1 should be subsets containing elements from Q[i]-th conjugacy class rep in each set but not any elements from j-th conj class for j < i */
until i eq #Q; 
return SetToIndexedSet(Total);
end function;


/* Note: Below you have to manually change the interval of counting from 100 for small groups to 10000 for larger ones */

function Find_All_Tuples(W, M: A := {1},  T := Rank(W) +1, Order := false); 
if Order eq true then;
FR := FreeGroup(T);
printf "Base Ring of W: %o \n", BaseRing(W);
X := {@@};
printf "Number of tuples to check is %o \n", #M;
error if #M lt 20000, "Numbers of tuples to check is less than 20000";
j := 1; m := #M div 20000;
t := Cputime();
while j in [1..m+1] do; 
M_1 := {@ M[i]: i in [(j-1)*20000 +1..Min(j*20000, #M)]  | IsIrreducible(GModule(FR, Setseq(M[i]))) and P(Setseq(M[i]), T-2: A := A) and Sub_Order(W,Setseq(M[i])) eq #W @};
for m in M_1 do;
u, v := P(Setseq(m), T-2: A := A); 
Include(~X, v);
end for;
printf "Found %o new tuples \n", #M_1;
printf "Checked upto index %o of %o, computation time so far: %o \n", Min(j*20000, #M), #M, Cputime(t); 
j := j+1;
end while;
else; 
FR := FreeGroup(T);
printf "Base Ring of W: %o \n", BaseRing(W);
X := {@@};
printf "Number of tuples to check is %o \n", #M;
error if #M lt 20000, "Numbers of tuples to check is less than 20000";
j := 1; m := #M div 20000;
t := Cputime();
while j in [1..m+1] do; 
M_1 := {@ M[i]: i in [(j-1)*20000 +1..Min(j*20000, #M)]  | IsIrreducible(GModule(FR, Setseq(M[i]))) and P(Setseq(M[i]), T-2: A := A) @};
for m in M_1 do;
u, v := P(Setseq(m), T-2: A := A); 
Include(~X, v);
end for;
printf "Found %o new tuples \n", #M_1;
printf "Checked upto index %o of %o, computation time so far: %o \n", Min(j*20000, #M), #M, Cputime(t); 
j := j+1;
end while;
end if;
return X;
end function;