InstallMethod( ClassPositionsOfCentre,
    "for a homogeneous list",
    [ IsHomogeneousList ],
    function( chi )
    local deg, mdeg, degsquare;

    deg:= chi[1];
    mdeg:= - deg;
    degsquare:= deg^2;

    return Filtered( [ 1 .. Length( chi ) ],
               i -> chi[i] = deg or chi[i] = mdeg or
                    ( IsCyc( chi[i] ) and IsCycInt( chi[i] )
                      and chi[i] * GaloisCyc( chi[i], -1 ) = degsquare ) );
    end );


ElementOfList:=function(L,K)
	local j;
	for j in [1..Length(L)] do
		if K=L[j] then
			return j;
		fi;
	od;
	return 0;
end;



prime_list:=function(i)
	local j, L;
	L:=[];
	for j in DivisorsInt(i) do
		if IsPrime(j) then 
			Add(L, j);
		fi;
	od;
	return L;
end;


list_max:=function(L)
	local i, K, S;
	K:=0;
	for i in [1..Length(L)] do
		if K<L[i] then 
			K:=L[i]; S:=i;
		fi;
	od;
	return [K, S];
end;

list_remoting:=function(L, K)
	local i, H;
	H:=[];
	for i in [1..Length(L)] do
		if i<K then Add(H, L[i]); else if i>K then Add(H, L[i]); fi; fi;		
	od; 
	return H;
end;

list_ordering:=function(L)
	local K, N,i;
	K:=[]; N:=L;
	for i in [1..Length(L)] do
		Add(K, list_max(N)[1]);
		N:=list_remoting(N,list_max(N)[2]);
		od;
	return K;
end;

		
Element_howmany_noncent_ConjugacyClasses:=function(G,K)
	local L, i, j;
	L:=list_ordering(SizesConjugacyClasses(CharacterTable(G)));
	j:=0;
	if K>0 then
	for i in [1..Length(L)] do
		j:=j+L[i];
		if j>=K then
			return i;
		fi;
	od;
	else return 0;
	fi;
end;

gallagher_g:=function(G)
	return List(Irr(G), f->(DegreeOfCharacter(f)^2-1)*Size(CentreOfCharacter(f)));
end;


list_appering:=function(L, K)
	local i;
	for i in L do
		if ElementOfList(K, i)>0 then else return 1; fi;
	od;
	return 0;
end;

gallagher_c:=function(G)
	local f, L,s;
	L:=[];
	for f in Irr(G) do
		s:=Element_howmany_noncent_ConjugacyClasses(G,(DegreeOfCharacter(f)^2-1)*Size(CentreOfCharacter(f)));
		Add(L,s);
	od;
	return L;
end;

ConjugacyClasses_howmany_Element:=function(G, K)
	local L, i, j, C;
	C:=Size(Centre(G));
	L:=list_ordering(SizesConjugacyClasses(CharacterTable(G)));
	j:=0;
	for i in [C..C+K-1] do
		j:=j+L[Length(L)-i];
	od;
	return j;
end;	

chillag_c:=function(G)
	local f, i, C, c, K, H;
	C:=[];
	if DerivedSubgroup(G)=G then else
		K:=CharacterTable(G);
		for f in Irr(G) do
			c:=1;
			if DegreeOfCharacter(f)>1 then
				H:=DerivedSubgroup(G);
				if IsIrreducibleCharacter(CharacterTable(H),RestrictedClassFunction(K,f,H)) then c:=2; else
					c:=Size(G);
					for i in [1..NrConjugacyClasses(G)] do
						if f[i]=0 then  c:=Minimum(c,SizesCentralizers(K)[i]);
						fi;
					od;
				fi; 
				else c:=0;
			fi;
			if IsInt(c/2) then Add(C,c/2); else Add(C,(c+1)/2);
			fi;
		od;
	fi;
	return C;
end;

chillag_g:=function(G)
	local L,C,i,g, H,KH,KG ;
	L:=[];
	if IsInt(Size(G)/2) then g:=Size(G)/2; else g:=(Size(G)+1)/2; fi;
	C:=chillag_c(G);
	H:=DerivedSubgroup(G); KH:=CharacterTable(H);
	KG:=CharacterTable(G);
	for i in [1..Length(C)] do
		if C[i]=1 then 
			if IsIrreducibleCharacter(KH,RestrictedClassFunction(KG,Irr(G)[i],H)) 
				then Add(L, ConjugacyClasses_howmany_Element(G,1));
				else Add(L,g);
			fi; 
		else  Add(L, ConjugacyClasses_howmany_Element(G,C[i]));
		fi;
	od;
	return L;
end;

IsElementNeg:=function(L)
	local i;
	for i in L do
		if i<0 then return true; fi;
	od;
	return false;
end;


sqfr:=function(n)
	local L, i, V;
	L:=DivisorsInt(n);
	V:=1;
	for i in L do
		if IsPrime(i) then V:=V*i;
		fi;
	od;
	return V;
end;

wilde_c:=function(G)
	local f, g, k, n, s, L;
	L:=[];
	g:=Size(G);
	for f in Irr(G) do
		s:=0;
		if f[1]=1 then  else 	
		for k in ConjugacyClasses(G) do
			n:=Order(Representative(k));
			if IsInt(g*g/f[1]/f[1]/n/sqfr(n)) and IsInt(g*g*g/f[1]/f[1]/n/n/n) then else
				s:=s+1;
			fi;
		od;
		if s=0 then s:=1; fi;
		fi;
		Add(L,s);
	od;
	return(L);
end;


wilde_g:=function(G)
	local f, g, k, n, s, L;
	L:=[];
	g:=Size(G);
	for f in Irr(G) do
		s:=0; 
		if f[1]=1 then else 
		for k in ConjugacyClasses(G) do
			n:=Order(Representative(k));
			if IsInt(g*g/f[1]/f[1]/n/sqfr(n)) and IsInt(g*g*g/f[1]/f[1]/n/n/n) then else
				s:=s+Size(k);
			fi;
		od;
		if s=0 then s:=Minimum(list_remoting(Set(SizesConjugacyClasses(CharacterTable(G))),1)); fi; fi;
		Add(L,s);
	od;
	return(L);
end;



compare_Ga_Ch_c:=function(R)
	local i, j, G;
	for i in [1..R] do
		for j in [1..NrSmallGroups(i)] do
			G:=SmallGroup(i,j);
			if G=DerivedSubgroup(G) then else
			if IsElementNeg(gallagher_c(G)-chillag_c(G)) then Print([i,j]); return false; fi;
			fi;
		od;
	od;
	return true;
end;

compare_Ch_Ga_c:=function(R)
	local i, j, G;
	for i in [1..R] do
		for j in [1..NrSmallGroups(i)] do
			G:=SmallGroup(i,j);
			if G=DerivedSubgroup(G) then else
			if IsElementNeg(-gallagher_c(G)+chillag_c(G)) then Print([i,j]); return false; fi;
			fi;
		od;
	od;
	return true;
end;

compare_W_Ch_c:=function(R)
	local i, j, G;
	for i in [1..R] do
		for j in [1..NrSmallGroups(i)] do
			G:=SmallGroup(i,j);
			if G=DerivedSubgroup(G) then else
			if IsElementNeg(wilde_c(G)-chillag_c(G)) then Print([i,j]); return false; fi;
			fi;
		od;
	od;
	return true;
end;

compare_Ch_W_c:=function(R)
	local i, j, G;
	for i in [1..R] do
		for j in [1..NrSmallGroups(i)] do
			G:=SmallGroup(i,j);
			if G=DerivedSubgroup(G) then else
			if IsElementNeg(-wilde_c(G)+chillag_c(G)) then Print([i,j]); return false; fi;
			fi;
		od;
	od;
	return true;
end;

compare_W_Ga_c:=function(R)
	local i, j, G;
	for i in [1..R] do
		for j in [1..NrSmallGroups(i)] do
			G:=SmallGroup(i,j);
			if IsElementNeg(wilde_c(G)-gallagher_c(G)) then Print([i,j]); return false;
			fi;
		od;
	od;
	return true;
end;

compare_Ga_W_c:=function(R)
	local i, j, G;
	for i in [1..R] do
		for j in [1..NrSmallGroups(i)] do
			G:=SmallGroup(i,j);
			if IsElementNeg(-wilde_c(G)+gallagher_c(G)) then Print([i,j]); return false;
			fi;
		od;
	od;
	return true;
end;

compare_Ga_Ch_g:=function(R)
	local i, j, G;
	for i in [1..R] do
		for j in [1..NrSmallGroups(i)] do
			G:=SmallGroup(i,j);
			if G=DerivedSubgroup(G) then else
			if IsElementNeg(gallagher_g(G)-chillag_g(G)) then Print([i,j]); return false; fi;
			fi;
		od;
	od;
	return true;
end;

compare_Ch_Ga_g:=function(R)
	local i, j, G;
	for i in [1..R] do
		for j in [1..NrSmallGroups(i)] do
			G:=SmallGroup(i,j);
			if G=DerivedSubgroup(G) then else
			if IsElementNeg(-gallagher_g(G)+chillag_g(G)) then Print([i,j]); return false; fi;
			fi;
		od;
	od;
	return true;
end;

compare_W_Ch_g:=function(R)
	local i, j, G;
	for i in [1..R] do
		for j in [1..NrSmallGroups(i)] do
			G:=SmallGroup(i,j);
			if G=DerivedSubgroup(G) then else
			if IsElementNeg(wilde_g(G)-chillag_g(G)) then Print([i,j]); return false; fi;
			fi;
		od;
	od;
	return true;
end;

compare_Ch_W_g:=function(R)
	local i, j, G;
	for i in [1..R] do
		for j in [1..NrSmallGroups(i)] do
			G:=SmallGroup(i,j);
			if G=DerivedSubgroup(G) then else
			if IsElementNeg(-wilde_g(G)+chillag_g(G)) then Print([i,j]); return false; fi;
			fi;
		od;
	od;
	return true;
end;



compare_W_Ga_g:=function(R)
	local i, j, G;
	for i in [1..R] do
		for j in [1..NrSmallGroups(i)] do
			G:=SmallGroup(i,j);
			if IsElementNeg(wilde_g(G)-gallagher_g(G)) then Print([i,j]); return false; fi;
		od;
	od;
	return true;
end;

compare_Ga_W_g:=function(R)
	local i, j, G;
	for i in [1..R] do
		for j in [1..NrSmallGroups(i)] do
			G:=SmallGroup(i,j);
			if IsElementNeg(-wilde_g(G)+gallagher_g(G)) then Print([i,j]); return false; fi;
		od;
	od;
	return true;
end;

Print("compare_x_y returns  the ID of the smallest group on which  the estimate 'y' is greater than 'x'." ); Print("\n");