How can I remove the redundant generators in PermutationGroup?

Question

Consider:

g1 = PermutationGroup[{Cycles[{{1, 12}, {2, 11}, {3, 10}, {4, 9}, {5, 8}, {6, 7}}],
    Cycles[{{1, 10}, {2, 11}, {3, 12}, {4, 7}, {5, 8}, {6, 9}}],
    Cycles[{{1, 3}, {4, 6}, {7, 9}, {10, 12}}]}];
g2 = PermutationGroup[{Cycles[{{1, 12}, {2, 11}, {3, 10}, {4, 9}, {5, 8}, {6, 7}}],
    Cycles[{{1, 10}, {2, 11}, {3, 12}, {4, 7}, {5, 8}, {6, 9}}]}];
g1 == g2

True

This means there is a redundant generator in the group g1. Maple has a function, NonRedundantGenerators, to deal with this. I can get the group without a redundant generator in MMA like this:

PermutationGroup[SelectFirst[Subsets[GroupGenerators[g1]], PermutationGroup[#] == g1 &]]

PermutationGroup[{Cycles[{{1,12},{2,11},{3,10},{4,9},{5,8},{6,7}}],Cycles[{{1,10},{2,11},{3,12},{4,7},{5,8},{6,9}}]}]

But when there are too many generated elements, like this group:

G = PermutationGroup[{Cycles[{}],
    Cycles[{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
       17, 18, 19, 20}}],
    Cycles[{{1, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7,
       6, 5, 4, 3, 2}}],
    Cycles[{{1, 4, 7, 10, 13, 16, 19, 2, 5, 8, 11, 14, 17, 20, 3, 6,
       9, 12, 15, 18}}],
    Cycles[{{1, 18, 15, 12, 9, 6, 3, 20, 17, 14, 11, 8, 5, 2, 19, 16,
       13, 10, 7, 4}}],
    Cycles[{{1, 8, 15, 2, 9, 16, 3, 10, 17, 4, 11, 18, 5, 12, 19, 6,
       13, 20, 7, 14}}],
    Cycles[{{1, 14, 7, 20, 13, 6, 19, 12, 5, 18, 11, 4, 17, 10, 3, 16,
        9, 2, 15, 8}}],
    Cycles[{{1, 10, 19, 8, 17, 6, 15, 4, 13, 2, 11, 20, 9, 18, 7, 16,
       5, 14, 3, 12}}],
    Cycles[{{1, 12, 3, 14, 5, 16, 7, 18, 9, 20, 11, 2, 13, 4, 15, 6,
       17, 8, 19, 10}}],
    Cycles[{{1, 3, 5, 7, 9, 11, 13, 15, 17, 19}, {2, 4, 6, 8, 10, 12,
       14, 16, 18, 20}}],
    Cycles[{{1, 19, 17, 15, 13, 11, 9, 7, 5, 3}, {2, 20, 18, 16, 14,
       12, 10, 8, 6, 4}}],
    Cycles[{{1, 7, 13, 19, 5, 11, 17, 3, 9, 15}, {2, 8, 14, 20, 6, 12,
        18, 4, 10, 16}}],
    Cycles[{{1, 15, 9, 3, 17, 11, 5, 19, 13, 7}, {2, 16, 10, 4, 18,
       12, 6, 20, 14, 8}}],
    Cycles[{{1, 5, 9, 13, 17}, {2, 6, 10, 14, 18}, {3, 7, 11, 15,
       19}, {4, 8, 12, 16, 20}}],
    Cycles[{{1, 17, 13, 9, 5}, {2, 18, 14, 10, 6}, {3, 19, 15, 11,
       7}, {4, 20, 16, 12, 8}}],
    Cycles[{{1, 9, 17, 5, 13}, {2, 10, 18, 6, 14}, {3, 11, 19, 7,
       15}, {4, 12, 20, 8, 16}}],
    Cycles[{{1, 13, 5, 17, 9}, {2, 14, 6, 18, 10}, {3, 15, 7, 19,
       11}, {4, 16, 8, 20, 12}}],
    Cycles[{{1, 6, 11, 16}, {2, 7, 12, 17}, {3, 8, 13, 18}, {4, 9, 14,
        19}, {5, 10, 15, 20}}],
    Cycles[{{1, 16, 11, 6}, {2, 17, 12, 7}, {3, 18, 13, 8}, {4, 19,
       14, 9}, {5, 20, 15, 10}}],
    Cycles[{{1, 3}, {4, 20}, {5, 19}, {6, 18}, {7, 17}, {8, 16}, {9,
       15}, {10, 14}, {11, 13}}],
    Cycles[{{1, 5}, {2, 4}, {6, 20}, {7, 19}, {8, 18}, {9, 17}, {10,
       16}, {11, 15}, {12, 14}}],
    Cycles[{{1, 7}, {2, 6}, {3, 5}, {8, 20}, {9, 19}, {10, 18}, {11,
       17}, {12, 16}, {13, 15}}],
    Cycles[{{1, 9}, {2, 8}, {3, 7}, {4, 6}, {10, 20}, {11, 19}, {12,
       18}, {13, 17}, {14, 16}}],
    Cycles[{{1, 11}, {2, 10}, {3, 9}, {4, 8}, {5, 7}, {12, 20}, {13,
       19}, {14, 18}, {15, 17}}],
    Cycles[{{1, 13}, {2, 12}, {3, 11}, {4, 10}, {5, 9}, {6, 8}, {14,
       20}, {15, 19}, {16, 18}}],
    Cycles[{{1, 15}, {2, 14}, {3, 13}, {4, 12}, {5, 11}, {6, 10}, {7,
       9}, {16, 20}, {17, 19}}],
    Cycles[{{1, 17}, {2, 16}, {3, 15}, {4, 14}, {5, 13}, {6, 12}, {7,
       11}, {8, 10}, {18, 20}}],
    Cycles[{{1, 19}, {2, 18}, {3, 17}, {4, 16}, {5, 15}, {6, 14}, {7,
       13}, {8, 12}, {9, 11}}],
    Cycles[{{2, 20}, {3, 19}, {4, 18}, {5, 17}, {6, 16}, {7, 15}, {8,
       14}, {9, 13}, {10, 12}}],
    Cycles[{{1, 2}, {3, 20}, {4, 19}, {5, 18}, {6, 17}, {7, 16}, {8,
       15}, {9, 14}, {10, 13}, {11, 12}}],
    Cycles[{{1, 4}, {2, 3}, {5, 20}, {6, 19}, {7, 18}, {8, 17}, {9,
       16}, {10, 15}, {11, 14}, {12, 13}}],
    Cycles[{{1, 6}, {2, 5}, {3, 4}, {7, 20}, {8, 19}, {9, 18}, {10,
       17}, {11, 16}, {12, 15}, {13, 14}}],
    Cycles[{{1, 8}, {2, 7}, {3, 6}, {4, 5}, {9, 20}, {10, 19}, {11,
       18}, {12, 17}, {13, 16}, {14, 15}}],
    Cycles[{{1, 10}, {2, 9}, {3, 8}, {4, 7}, {5, 6}, {11, 20}, {12,
       19}, {13, 18}, {14, 17}, {15, 16}}],
    Cycles[{{1, 12}, {2, 11}, {3, 10}, {4, 9}, {5, 8}, {6, 7}, {13,
       20}, {14, 19}, {15, 18}, {16, 17}}],
    Cycles[{{1, 14}, {2, 13}, {3, 12}, {4, 11}, {5, 10}, {6, 9}, {7,
       8}, {15, 20}, {16, 19}, {17, 18}}],
    Cycles[{{1, 16}, {2, 15}, {3, 14}, {4, 13}, {5, 12}, {6, 11}, {7,
       10}, {8, 9}, {17, 20}, {18, 19}}],
    Cycles[{{1, 18}, {2, 17}, {3, 16}, {4, 15}, {5, 14}, {6, 13}, {7,
       12}, {8, 11}, {9, 10}, {19, 20}}],
    Cycles[{{1, 20}, {2, 19}, {3, 18}, {4, 17}, {5, 16}, {6, 15}, {7,
       14}, {8, 13}, {9, 12}, {10, 11}}],
    Cycles[{{1, 11}, {2, 12}, {3, 13}, {4, 14}, {5, 15}, {6, 16}, {7,
       17}, {8, 18}, {9, 19}, {10, 20}}]}];

this method does not work anymore. Is there another way?

Answer 1

Here is a quick and dirty attempt to solve your problem. delCycshould delete redundant cycles:

g1 = PermutationGroup[{Cycles[{{1, 12}, {2, 11}, {3, 10}, {4, 9}, {5, 
       8}, {6, 7}}], 
    Cycles[{{1, 10}, {2, 11}, {3, 12}, {4, 7}, {5, 8}, {6, 9}}], 
    Cycles[{{1, 3}, {4, 6}, {7, 9}, {10, 12}}]}];
g2 = PermutationGroup[{Cycles[{{1, 12}, {2, 11}, {3, 10}, {4, 9}, {5, 
       8}, {6, 7}}], 
    Cycles[{{1, 10}, {2, 11}, {3, 12}, {4, 7}, {5, 8}, {6, 9}}]}];

delCyc[g_] := Module[{t = g[[1]],tmp},
  If[PermutationGroup[t] == PermutationGroup[tmp = DeleteCases[t, #]],
      t = tmp] & /@ t;
  PermutationGroup[t]
  ]

To test if it works:

delCyc[g1]
(* PermutationGroup[{Cycles[{{1, 10}, {2, 11}, {3, 12}, {4, 7}, {5, 
     8}, {6, 9}}], Cycles[{{1, 3}, {4, 6}, {7, 9}, {10, 12}}]}] *)

And with your large example G:

delCyc[G]
(* PermutationGroup[{Cycles[{{2, 20}, {3, 19}, {4, 18}, {5, 17}, {6, 
     16}, {7, 15}, {8, 14}, {9, 13}, {10, 12}}], 
  Cycles[{{1, 20}, {2, 19}, {3, 18}, {4, 17}, {5, 16}, {6, 15}, {7, 
     14}, {8, 13}, {9, 12}, {10, 11}}]}] *)