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# This function computes the SC-table for
# the cohomology ring. Unlike other functions,
# this one is not idempotent because of the way
# the SC-table is set up.
InstallGlobalFunction(MakeMultiplicationTable,function(C,n)
local G,K,R,L,B,N,T,h,d,c,i,j,k,l,BB,s,t,u,v,w,x;
G:=C!.G; K:=C!.K;
R:=C!.R; L:=C!.L; B:=C!.B;
# Set up the SC table.
N:=Sum(B{[1..n+1]}); # Size of the table
C!.T:=EmptySCTable(N,Zero(K));
T:=C!.T;
# Set up the identity element of of the table.
for i in [1..N] do
T[1][i]:=[[i],[One(K)]];
T[i][1]:=[[i],[One(K)]];
od;
# Create all cochains.
C!.h:=[]; h:=C!.h;
for j in [1..n] do
h[j]:=List(IdentityMat(B[j+1],K),x->FirstLift(C,x));
od;
# Lift maps and record products.
for i in [1..n-1] do
for j in [1..Minimum(i,n-i)] do
BB:=Basis(FullRowSpace(K,B[i+j+1]));
for k in [1..B[i+1]] do
LiftChainMap(C,h[i][k],i,j);
for l in [1..B[j+1]] do
v:=ExtractColumn(h[i][k][j+1],(l-1)*Size(G)+1);
# s and t are the positions of the elements
# we're multiplying.
s:=Sum(B{[1..i]})+k;
t:=Sum(B{[1..j]})+l;
# The following funny-looking lines just prepare
# the (t,s) and (s,t) entries of SC table.
u:=List([1..Sum(B{[1..i+j]})],x->Zero(K));
Append(u,Coefficients(BB,v));
w:=TransposedMat(ListX([1..Size(u)],
x->u[x]<>Zero(K), x->[x,u[x]]));
if w=[] then w:=[[],[]];fi;
T[s][t]:=w;
if i<>j then
x:=ShallowCopy(w);
x[2]:=(-1)^(i*j)*x[2];
T[t][s]:=x;
fi;
od;
od;
od;
od;
end
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