Permutations[Range[12]] produces an error instead of a list

Since Mathematica 8 it is possible generate the elements of any group one by one with GroupElements. Here's for example a randomly chosen element of the permutation group on 20 elements:

GroupElements[SymmetricGroup[20], {10^6 + 1}]

{Cycles[{{11, 13, 19}, {12, 18, 17, 16}, {15, 20}}]}

The result is immediately; there's no need to build up the full group of $20! \approx 2.4 \cdot 10^{18}$ elements. We can use this to scan over the permutations of a given list without generating them all at once:

ScanPermutations[function_, list_] /; Length[list] <= 8 :=
  Scan[function, Permutations @ list];

ScanPermutations[function_, list_] :=
  Do[
    Scan[
      function @ Permute[list, #] &,
      GroupElements[
        SymmetricGroup @ Length @ list,
        Range[8! * (i - 1) + 1, 8! * i]
      ]
    ],
    {i, Length[list]! / 8!}
  ];

The code splits up the permutations in blocks of $8! = 40320$ whenever the given list has more than 8 elements. This is about ten times faster than calling GroupElements to generate just one permutation at a time.

Printing e.g. the permutations of {1,2,3} can now be done as follows:

ScanPermutations[Print, {1,2,3}]

{1,2,3}
{1,3,2}
{2,1,3}
{3,1,2}
{2,3,1}
{3,2,1}

Going one step further, we may use Reap and Sow to select permutations that match a given criterion:

SelectPermutations[list_, crit_] :=
  First @ Last @ Reap @ ScanPermutations[
    If[crit @ #, Sow @ #] &,
    list
  ];

And finally, here's a small example that puts the above in action. It selects the even permutations of {1,2,3}:

SelectPermutations[{1, 2, 3}, Signature[#] === 1 &]

{{1, 2, 3}, {2, 3, 1}, {3, 1, 2}}

Combinatorica` has the function NextPermutation which allows you to iterate over the permutations. There may be ways of generating a smaller subset if you have more information about what you are looking for.

Consider than the permutations of {1, 2, 3, 4, 5} are each of the permutations of {1, 2, 3, 4} with 5 inserted at each possible place. One can therefore examine the permutations of {1, 2, 3, 4, 5} in blocks like this:

p4 = Permutations@Range@4;

Table[
  ReplaceList[x, {h___, t___} :> {h, 5, t}],
  {x, p4}
]

For example, making a certain selection:

Table[
  Select[
    ReplaceList[x, {h___, t___} :> {h, 5, t}],
    # + #2 - #3*#4/#5 > 7 & @@ # &
  ],
  {x, p4}
]

The same can be applied to the permutations of Range@12.