code: def double_coset_can_rep(dummies, sym, b_S, sgens, S_transversals, g):
      r"""
      Butler-Portugal algorithm for tensor canonicalization with dummy indices.
  
      Parameters
      ==========
  
        dummies
          list of lists of dummy indices,
          one list for each type of index;
          the dummy indices are put in order contravariant, covariant
          [d0, -d0, d1, -d1, ...].
  
        sym
          list of the symmetries of the index metric for each type.
  
        possible symmetries of the metrics
                * 0     symmetric
                * 1     antisymmetric
                * None  no symmetry
  
        b_S
          base of a minimal slot symmetry BSGS.
  
        sgens
          generators of the slot symmetry BSGS.
  
        S_transversals
          transversals for the slot BSGS.
  
        g
          permutation representing the tensor.
  
      Returns
      =======
  
      Return 0 if the tensor is zero, else return the array form of
      the permutation representing the canonical form of the tensor.
  
      Notes
      =====
  
      A tensor with dummy indices can be represented in a number
      of equivalent ways which typically grows exponentially with
      the number of indices. To be able to establish if two tensors
      with many indices are equal becomes computationally very slow
      in absence of an efficient algorithm.
  
      The Butler-Portugal algorithm [3] is an efficient algorithm to
      put tensors in canonical form, solving the above problem.
  
      Portugal observed that a tensor can be represented by a permutation,
      and that the class of tensors equivalent to it under slot and dummy
      symmetries is equivalent to the double coset `D*g*S`
      (Note: in this documentation we use the conventions for multiplication
      of permutations p, q with (p*q)(i) = p[q[i]] which is opposite
      to the one used in the Permutation class)
  
      Using the algorithm by Butler to find a representative of the
      double coset one can find a canonical form for the tensor.
  
      To see this correspondence,
      let `g` be a permutation in array form; a tensor with indices `ind`
      (the indices including both the contravariant and the covariant ones)
      can be written as
  
      `t = T(ind[g[0]], \dots, ind[g[n-1]])`,
  
      where `n = len(ind)`;
      `g` has size `n + 2`, the last two indices for the sign of the tensor
      (trick introduced in [4]).
  
      A slot symmetry transformation `s` is a permutation acting on the slots
      `t \rightarrow T(ind[(g*s)[0]], \dots, ind[(g*s)[n-1]])`
  
      A dummy symmetry transformation acts on `ind`
      `t \rightarrow T(ind[(d*g)[0]], \dots, ind[(d*g)[n-1]])`
  
      Being interested only in the transformations of the tensor under
      these symmetries, one can represent the tensor by `g`, which transforms
      as
  
      `g -> d*g*s`, so it belongs to the coset `D*g*S`, or in other words
      to the set of all permutations allowed by the slot and dummy symmetries.
  
      Let us explain the conventions by an example.
  
      Given a tensor `T^{d3 d2 d1}{}_{d1 d2 d3}` with the slot symmetries
            `T^{a0 a1 a2 a3 a4 a5} = -T^{a2 a1 a0 a3 a4 a5}`
  
            `T^{a0 a1 a2 a3 a4 a5} = -T^{a4 a1 a2 a3 a0 a5}`
  
      and symmetric metric, find the tensor equivalent to it which
      is the lowest under the ordering of indices:
      lexicographic ordering `d1, d2, d3` and then contravariant
      before covariant index; that is the canonical form of the tensor.
  
      The canonical form is `-T^{d1 d2 d3}{}_{d1 d2 d3}`
      obtained using `T^{a0 a1 a2 a3 a4 a5} = -T^{a2 a1 a0 a3 a4 a5}`.
  
      To convert this problem in the input for this function,
      use the following ordering of the index names
      (- for covariant for short) `d1, -d1, d2, -d2, d3, -d3`
  
      `T^{d3 d2 d1}{}_{d1 d2 d3}` corresponds to `g = [4, 2, 0, 1, 3, 5, 6, 7]`
      where the last two indices are for the sign
  
      `sgens = [Permutation(0, 2)(6, 7), Permutation(0, 4)(6, 7)]`
  
      sgens[0] is the slot symmetry `-(0, 2)`
      `T^{a0 a1 a2 a3 a4 a5} = -T^{a2 a1 a0 a3 a4 a5}`
  
      sgens[1] is the slot symmetry `-(0, 4)`
      `T^{a0 a1 a2 a3 a4 a5} = -T^{a4 a1 a2 a3 a0 a5}`
  
      The dummy symmetry group D is generated by the strong base generators
      `[(0, 1), (2, 3), (4, 5), (0, 2)(1, 3), (0, 4)(1, 5)]`
      where the first three interchange covariant and contravariant
      positions of the same index (d1 <-> -d1) and the last two interchange
      the dummy indices themselves (d1 <-> d2).
  
      The dummy symmetry acts from the left
      `d = [1, 0, 2, 3, 4, 5, 6, 7]`  exchange `d1 \leftrightarrow -d1`
      `T^{d3 d2 d1}{}_{d1 d2 d3} == T^{d3 d2}{}_{d1}{}^{d1}{}_{d2 d3}`
  
      `g=[4, 2, 0, 1, 3, 5, 6, 7]  -> [4, 2, 1, 0, 3, 5, 6, 7] = _af_rmul(d, g)`
      which differs from `_af_rmul(g, d)`.
  
      The slot symmetry acts from the right
      `s = [2, 1, 0, 3, 4, 5, 7, 6]`  exchanges slots 0 and 2 and changes sign
      `T^{d3 d2 d1}{}_{d1 d2 d3} == -T^{d1 d2 d3}{}_{d1 d2 d3}`
  
      `g=[4,2,0,1,3,5,6,7]  -> [0, 2, 4, 1, 3, 5, 7, 6] = _af_rmul(g, s)`
  
      Example in which the tensor is zero, same slot symmetries as above:
      `T^{d2}{}_{d1 d3}{}^{d1 d3}{}_{d2}`
  
      `= -T^{d3}{}_{d1 d3}{}^{d1 d2}{}_{d2}`   under slot symmetry `-(0,4)`;
  
      `= T_{d3 d1}{}^{d3}{}^{d1 d2}{}_{d2}`    under slot symmetry `-(0,2)`;
  
      `= T^{d3}{}_{d1 d3}{}^{d1 d2}{}_{d2}`    symmetric metric;
  
      `= 0`  since two of these lines have tensors differ only for the sign.
  
      The double coset D*g*S consists of permutations `h = d*g*s` corresponding
      to equivalent tensors; if there are two `h` which are the same apart
      from the sign, return zero; otherwise
      choose as representative the tensor with indices
      ordered lexicographically according to `[d1, -d1, d2, -d2, d3, -d3]`
      that is ``rep = min(D*g*S) = min([d*g*s for d in D for s in S])``
  
      The indices are fixed one by one; first choose the lowest index
      for slot 0, then the lowest remaining index for slot 1, etc.
      Doing this one obtains a chain of stabilizers
  
      `S \rightarrow S_{b0} \rightarrow S_{b0,b1} \rightarrow \dots` and
      `D \rightarrow D_{p0} \rightarrow D_{p0,p1} \rightarrow \dots`
  
      where ``[b0, b1, ...] = range(b)`` is a base of the symmetric group;
      the strong base `b_S` of S is an ordered sublist of it;
      therefore it is sufficient to compute once the
      strong base generators of S using the Schreier-Sims algorithm;
      the stabilizers of the strong base generators are the
      strong base generators of the stabilizer subgroup.
  
      ``dbase = [p0, p1, ...]`` is not in general in lexicographic order,
      so that one must recompute the strong base generators each time;
      however this is trivial, there is no need to use the Schreier-Sims
      algorithm for D.
  
      The algorithm keeps a TAB of elements `(s_i, d_i, h_i)`
      where `h_i = d_i \times g \times s_i` satisfying `h_i[j] = p_j` for `0 \le j < i`
      starting from `s_0 = id, d_0 = id, h_0 = g`.
  
      The equations `h_0[0] = p_0, h_1[1] = p_1, \dots` are solved in this order,
      choosing each time the lowest possible value of p_i
  
      For `j < i`
      `d_i*g*s_i*S_{b_0, \dots, b_{i-1}}*b_j = D_{p_0, \dots, p_{i-1}}*p_j`
      so that for dx in `D_{p_0,\dots,p_{i-1}}` and sx in
      `S_{base[0], \dots, base[i-1]}` one has `dx*d_i*g*s_i*sx*b_j = p_j`
  
      Search for dx, sx such that this equation holds for `j = i`;
      it can be written as `s_i*sx*b_j = J, dx*d_i*g*J = p_j`
      `sx*b_j = s_i**-1*J; sx = trace(s_i**-1, S_{b_0,...,b_{i-1}})`
      `dx**-1*p_j = d_i*g*J; dx = trace(d_i*g*J, D_{p_0,...,p_{i-1}})`
  
      `s_{i+1} = s_i*trace(s_i**-1*J, S_{b_0,...,b_{i-1}})`
      `d_{i+1} = trace(d_i*g*J, D_{p_0,...,p_{i-1}})**-1*d_i`
      `h_{i+1}*b_i = d_{i+1}*g*s_{i+1}*b_i = p_i`
  
      `h_n*b_j = p_j` for all j, so that `h_n` is the solution.
  
      Add the found `(s, d, h)` to TAB1.
  
      At the end of the iteration sort TAB1 with respect to the `h`;
      if there are two consecutive `h` in TAB1 which differ only for the
      sign, the tensor is zero, so return 0;
      if there are two consecutive `h` which are equal, keep only one.
  
      Then stabilize the slot generators under `i` and the dummy generators
      under `p_i`.
  
      Assign `TAB = TAB1` at the end of the iteration step.
  
      At the end `TAB` contains a unique `(s, d, h)`, since all the slots
      of the tensor `h` have been fixed to have the minimum value according
      to the symmetries. The algorithm returns `h`.
  
      It is important that the slot BSGS has lexicographic minimal base,
      otherwise there is an `i` which does not belong to the slot base
      for which `p_i` is fixed by the dummy symmetry only, while `i`
      is not invariant from the slot stabilizer, so `p_i` is not in
      general the minimal value.
  
      This algorithm differs slightly from the original algorithm [3]:
        the canonical form is minimal lexicographically, and
        the BSGS has minimal base under lexicographic order.
        Equal tensors `h` are eliminated from TAB.
  
  
      Examples
      ========
  
      >>> from sympy.combinatorics.permutations import Permutation
      >>> from sympy.combinatorics.tensor_can import double_coset_can_rep, get_transversals
      >>> gens = [Permutation(x) for x in [[2, 1, 0, 3, 4, 5, 7, 6], [4, 1, 2, 3, 0, 5, 7, 6]]]
      >>> base = [0, 2]
      >>> g = Permutation([4, 2, 0, 1, 3, 5, 6, 7])
      >>> transversals = get_transversals(base, gens)
      >>> double_coset_can_rep([list(range(6))], [0], base, gens, transversals, g)
      [0, 1, 2, 3, 4, 5, 7, 6]
  
      >>> g = Permutation([4, 1, 3, 0, 5, 2, 6, 7])
      >>> double_coset_can_rep([list(range(6))], [0], base, gens, transversals, g)
      0
      """
      size = g.size
      g = g.array_form
      num_dummies = size - 2
      indices = list(range(num_dummies))
      all_metrics_with_sym = not any(_ is None for _ in sym)
      num_types = len(sym)
      dumx = dummies[:]
      dumx_flat = []
      for dx in dumx:
          dumx_flat.extend(dx)
      b_S = b_S[:]
      sgensx = [h._array_form for h in sgens]
      if b_S:
          S_transversals = transversal2coset(size, b_S, S_transversals)
      # strong generating set for D
      dsgsx = []
      for i in range(num_types):
          dsgsx.extend(dummy_sgs(dumx[i], sym[i], num_dummies))
      idn = list(range(size))
      # TAB = list of entries (s, d, h) where h = _af_rmuln(d,g,s)
      # for short, in the following d*g*s means _af_rmuln(d,g,s)
      TAB = [(idn, idn, g)]
      for i in range(size - 2):
          b = i
          testb = b in b_S and sgensx
          if testb:
              sgensx1 = [_af_new(_) for _ in sgensx]
              deltab = _orbit(size, sgensx1, b)
          else:
              deltab = {b}
          # p1 = min(IMAGES) = min(Union D_p*h*deltab for h in TAB)
          if all_metrics_with_sym:
              md = _min_dummies(dumx, sym, indices)
          else:
              md = [
                  min(_orbit(size, [_af_new(ddx) for ddx in dsgsx], ii))
                  for ii in range(size - 2)
              ]
  
          p_i = min(
              [min([md[h[x]] for x in deltab]) for s, d, h in TAB]
          )
          dsgsx1 = [_af_new(_) for _ in dsgsx]
          Dxtrav = (
              _orbit_transversal(size, dsgsx1, p_i, False, af=True)
              if dsgsx
              else None
          )
          if Dxtrav:
              Dxtrav = [_af_invert(x) for x in Dxtrav]
          # compute the orbit of p_i
          for ii in range(num_types):
              if p_i in dumx[ii]:
                  # the orbit is made by all the indices in dum[ii]
                  if sym[ii] is not None:
                      deltap = dumx[ii]
                  else:
                      # the orbit is made by all the even indices if p_i
                      # is even, by all the odd indices if p_i is odd
                      p_i_index = dumx[ii].index(p_i) % 2
                      deltap = dumx[ii][p_i_index::2]
                  break
          else:
              deltap = [p_i]
          TAB1 = []
          while TAB:
              s, d, h = TAB.pop()
              if min([md[h[x]] for x in deltab]) != p_i:
                  continue
              deltab1 = [x for x in deltab if md[h[x]] == p_i]
              # NEXT = s*deltab1 intersection (d*g)**-1*deltap
              dg = _af_rmul(d, g)
              dginv = _af_invert(dg)
              sdeltab = [s[x] for x in deltab1]
              gdeltap = [dginv[x] for x in deltap]
              NEXT = [x for x in sdeltab if x in gdeltap]
              # d, s satisfy
              # d*g*s*base[i-1] = p_{i-1}; using the stabilizers
              # d*g*s*S_{base[0],...,base[i-1]}*base[i-1] =
              # D_{p_0,...,p_{i-1}}*p_{i-1}
              # so that to find d1, s1 satisfying d1*g*s1*b = p_i
              # one can look for dx in D_{p_0,...,p_{i-1}} and
              # sx in S_{base[0],...,base[i-1]}
              # d1 = dx*d; s1 = s*sx
              # d1*g*s1*b = dx*d*g*s*sx*b = p_i
              for j in NEXT:
                  if testb:
                      # solve s1*b = j with s1 = s*sx for some element sx
                      # of the stabilizer of ..., base[i-1]
                      # sx*b = s**-1*j; sx = _trace_S(s, j,...)
                      # s1 = s*trace_S(s**-1*j,...)
                      s1 = _trace_S(s, j, b, S_transversals)
                      if not s1:
                          continue
                      else:
                          s1 = [s[ix] for ix in s1]
                  else:
                      s1 = s
                  # assert s1[b] == j  # invariant
                  # solve d1*g*j = p_i with d1 = dx*d for some element dg
                  # of the stabilizer of ..., p_{i-1}
                  # dx**-1*p_i = d*g*j; dx**-1 = trace_D(d*g*j,...)
                  # d1 = trace_D(d*g*j,...)**-1*d
                  # to save an inversion in the inner loop; notice we did
                  # Dxtrav = [perm_af_invert(x) for x in Dxtrav] out of the loop
                  if Dxtrav:
                      d1 = _trace_D(dg[j], p_i, Dxtrav)
                      if not d1:
                          continue
                  else:
                      if p_i != dg[j]:
                          continue
                      d1 = idn
                  assert d1[dg[j]] == p_i  # invariant
                  d1 = [d1[ix] for ix in d]
                  h1 = [d1[g[ix]] for ix in s1]
                  # assert h1[b] == p_i  # invariant
                  TAB1.append((s1, d1, h1))
  
          # if TAB contains equal permutations, keep only one of them;
          # if TAB contains equal permutations up to the sign, return 0
          TAB1.sort(key=lambda x: x[-1])
          prev = [0] * size
          while TAB1:
              s, d, h = TAB1.pop()
              if h[:-2] == prev[:-2]:
                  if h[-1] != prev[-1]:
                      return 0
              else:
                  TAB.append((s, d, h))
              prev = h
  
          # stabilize the SGS
          sgensx = [h for h in sgensx if h[b] == b]
          if b in b_S:
              b_S.remove(b)
          _dumx_remove(dumx, dumx_flat, p_i)
          dsgsx = []
          for i in range(num_types):
              dsgsx.extend(dummy_sgs(dumx[i], sym[i], num_dummies))
      return TAB[0][-1]
  
imports: from sympy.combinatorics.permutations import Permutation, _af_rmul, _af_invert, _af_new
  from sympy.combinatorics.perm_groups import PermutationGroup, _orbit, _orbit_transversal
  from sympy.combinatorics.util import _distribute_gens_by_base, _orbits_transversals_from_bsgs
label_desc: Use image node_id 7 for calling a global function with example usage: double_coset_can_rep(dummies, sym, b_S, sgens, S_transversals, g) and returns: TAB, int