python3 - <<'EOF'
import itertools
def det(A):
 n=len(A)
 A=[list(r) for r in A]
 for c in range(n):
  p=next((r for r in range(c,n) if A[r][c]),None)
  if p is None:return 0
  A[c],A[p]=A[p],A[c]
  for r in range(c+1,n):
   if A[r][c]:A[r]=[x^y for x,y in zip(A[r],A[c])]
 return 1
def direct(n):
 pairs=[(i,j) for i in range(n) for j in range(i+1,n)]
 cnt=0
 for bits in itertools.product([0,1],repeat=len(pairs)):
  C=[[0]*n for _ in range(n)]
  C[0][0]=C[1][1]=1
  for (i,j),b in zip(pairs,bits):C[i][j]=C[j][i]=b
  if not det(C):continue
  ok=True
  for i in range(n):
   rows=C[:i]+C[i+1:]
   sub=[row[:i]+row[i+1:] for row in rows]
   if not det(sub):
    ok=False
    break
  if not ok:continue
  D=[[C[i][j]^(i==j) for j in range(n)] for i in range(n)]
  if det(D):cnt+=1
 return cnt
print('direct4',direct(4))
print('direct6',direct(6))
EOF