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Generator gen1: Rotattion of C 1 => 3 => 8 => 6 => 1\nWe form the set \ of the pairs {x,y} which describe the permutation (mapping) x \[Rule] y:"], "Text"], Cell[CellGroupData[{ Cell["\<\ l = {{1, 3}, {2, 5}, {3, 8}, {4, 2}, {5, 7}, {6, 1}, {7, 4}, {8, 6}, {9, 48}, \ {10, 47}, {11, 46}, {12, 9}, {13, 10}, {14, 11}, {15, 12}, {16, 13}, {17, 14}, {18, 18}, {19, 19}, {20, 20}, {21, \ 21}, {22, 22}, {23, 23}, {24, 24}, {25, 25}, {26, 26}, {27, 27}, {28, 28}, {29, 29}, {30, 30}, {31, 31}, {32, \ 32}, {33, 33}, {34, 34}, {35, 35}, {36, 36}, {37, 37}, {38, 38}, {39, 39}, {40, 40}, {41, 41}, {42, 42}, {43, \ 43}, {44, 44}, {45, 45}, {46, 17}, {47, 16}, {48, 15}}\ \>", "Input"], Cell[OutputFormData["\<\ {{1, 3}, {2, 5}, {3, 8}, {4, 2}, {5, 7}, {6, 1}, {7, 4}, {8, 6}, {9, 48}, \ {10, 47}, {11, 46}, {12, 9}, {13, 10}, {14, 11}, {15, 12}, {16, 13}, {17, 14}, {18, 18}, {19, 19}, {20, \ 20}, {21, 21}, {22, 22}, {23, 23}, {24, 24}, {25, 25}, {26, 26}, {27, 27}, {28, 28}, {29, 29}, {30, \ 30}, {31, 31}, {32, 32}, {33, 33}, {34, 34}, {35, 35}, {36, 36}, {37, 37}, {38, 38}, {39, 39}, {40, \ 40}, {41, 41}, {42, 42}, {43, 43}, {44, 44}, {45, 45}, {46, 17}, {47, 16}, {48, 15}}\ \>", "\<\ {{1, 3}, {2, 5}, {3, 8}, {4, 2}, {5, 7}, {6, 1}, {7, 4}, {8, 6}, {9, 48}, \ {10, 47}, {11, 46}, {12, 9}, {13, 10}, {14, 11}, {15, 12}, {16, 13}, {17, 14}, {18, 18}, {19, 19}, {20, \ 20}, {21, 21}, {22, 22}, {23, 23}, {24, 24}, {25, 25}, {26, 26}, {27, 27}, {28, 28}, {29, 29}, {30, \ 30}, {31, 31}, {32, 32}, {33, 33}, {34, 34}, {35, 35}, {36, 36}, {37, 37}, {38, 38}, {39, 39}, {40, \ 40}, {41, 41}, {42, 42}, {43, 43}, {44, 44}, {45, 45}, {46, 17}, {47, 16}, {48, 15}}\ \>"], "Output"] }, Open ]], Cell["Now we transform this set into a permutation.", "Text"], Cell[CellGroupData[{ Cell["gen1 = Perm @@ Transpose[l][[2]]", "Input"], Cell[OutputFormData["\<\ Perm[3, 5, 8, 2, 7, 1, 4, 6, 48, 47, 46, 9, 10, 11, 12, 13, 14, 18, 19, 20, \ 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 17, \ 16, 15]\ \>", "\<\ ( 3 5 8 2 7 1 4 6 48 47 46 9 10 11 12 13 14 18 19 20 21 22 23 24 25 26 27 28 \ 29 30 31 32 33 34 35 36 37\\ 38 39 40 41 42 43 44 45 17 16 15 )\ \>"], "Output"] }, Open ]], Cell["Generator gen2: Rotation of E 12 => 14 => 29 => 27 => 12", "Text"], Cell[CellGroupData[{ Cell["\<\ l = {{1, 1}, {2, 2}, {3, 3}, {4, 4}, {5, 5}, {6, 15}, {7, 22}, {8, 30}, {9, \ 9}, {10, 10}, {11, 8}, {12, 14}, {13, 21}, {14, 29}, {15, 35}, {16, 16}, {17, 17}, {18, 18}, {19, 7}, {20, 13}, {21, \ 28}, {22, 34}, {23, 23}, {24, 24}, {25, 25}, {26, 6}, {27, 12}, {28, 20}, {29, 27}, {30, 33}, {31, 31}, {32, \ 32}, {33, 11}, {34, 19}, {35, 26}, {36, 36}, {37, 37}, {38, 38}, {39, 39}, {40, 40}, {41, 41}, {42, 42}, {43, \ 43}, {44, 44}, {45, 45}, {46, 46}, {47, 47}, {48, 48}}\ \>", "Input"], Cell[OutputFormData["\<\ {{1, 1}, {2, 2}, {3, 3}, {4, 4}, {5, 5}, {6, 15}, {7, 22}, {8, 30}, {9, 9}, \ {10, 10}, {11, 8}, {12, 14}, {13, 21}, {14, 29}, {15, 35}, {16, 16}, {17, 17}, {18, 18}, {19, 7}, {20, \ 13}, {21, 28}, {22, 34}, {23, 23}, {24, 24}, {25, 25}, {26, 6}, {27, 12}, {28, 20}, {29, 27}, {30, \ 33}, {31, 31}, {32, 32}, {33, 11}, {34, 19}, {35, 26}, {36, 36}, {37, 37}, {38, 38}, {39, 39}, {40, \ 40}, {41, 41}, {42, 42}, {43, 43}, {44, 44}, {45, 45}, {46, 46}, {47, 47}, {48, 48}}\ \>", "\<\ {{1, 1}, {2, 2}, {3, 3}, {4, 4}, {5, 5}, {6, 15}, {7, 22}, {8, 30}, {9, 9}, \ {10, 10}, {11, 8}, {12, 14}, {13, 21}, {14, 29}, {15, 35}, {16, 16}, {17, 17}, {18, 18}, {19, 7}, {20, \ 13}, {21, 28}, {22, 34}, {23, 23}, {24, 24}, {25, 25}, {26, 6}, {27, 12}, {28, 20}, {29, 27}, {30, \ 33}, {31, 31}, {32, 32}, {33, 11}, {34, 19}, {35, 26}, {36, 36}, {37, 37}, {38, 38}, {39, 39}, {40, \ 40}, {41, 41}, {42, 42}, {43, 43}, {44, 44}, {45, 45}, {46, 46}, {47, 47}, {48, 48}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["gen2 = Perm @@ Transpose[l][[2]]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4, 5, 15, 22, 30, 9, 10, 8, 14, 21, 29, 35, 16, 17, 18, 7, 13, \ 28, 34, 23, 24, 25, 6, 12, 20, 27, 33, 31, 32, 11, 19, 26, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, \ 47, 48]\ \>", "\<\ ( 1 2 3 4 5 15 22 30 9 10 8 14 21 29 35 16 17 18 7 13 28 34 23 24 25 6 12 20 \ 27 33 31 32 11 19 26 36 37\\ 38 39 40 41 42 43 44 45 46 47 48 )\ \>"], "Output"] }, Open ]], Cell["Generator gen3: Rotation of B 9 => 11 => 26 => 24 => 9", "Text"], Cell[CellGroupData[{ Cell["\<\ l = {{1, 12}, {2, 2}, {3, 3}, {4, 20}, {5, 5}, {6, 27}, {7, 7}, {8, 8}, {9, \ 11}, {10, 19}, {11, 26}, {12, 33}, {13, 13}, {14, 14}, {15, 15}, {16, 16}, {17, 17}, {18, 10}, {19, 25}, {20, 36}, {21, \ 21}, {22, 22}, {23, 23}, {24, 9}, {25, 18}, {26, 24}, {27, 38}, {28, 28}, {29, 29}, {30, 30}, {31, 31}, {32, \ 32}, {33, 41}, {34, 34}, {35, 35}, {36, 44}, {37, 37}, {38, 46}, {39, 39}, {40, 40}, {41, 1}, {42, 42}, {43, \ 43}, {44, 4}, {45, 45}, {46, 6}, {47, 47}, {48, 48}}\ \>", "Input"], Cell[OutputFormData["\<\ {{1, 12}, {2, 2}, {3, 3}, {4, 20}, {5, 5}, {6, 27}, {7, 7}, {8, 8}, {9, 11}, \ {10, 19}, {11, 26}, {12, 33}, {13, 13}, {14, 14}, {15, 15}, {16, 16}, {17, 17}, {18, 10}, {19, \ 25}, {20, 36}, {21, 21}, {22, 22}, {23, 23}, {24, 9}, {25, 18}, {26, 24}, {27, 38}, {28, 28}, {29, \ 29}, {30, 30}, {31, 31}, {32, 32}, {33, 41}, {34, 34}, {35, 35}, {36, 44}, {37, 37}, {38, 46}, {39, \ 39}, {40, 40}, {41, 1}, {42, 42}, {43, 43}, {44, 4}, {45, 45}, {46, 6}, {47, 47}, {48, 48}}\ \>", "\<\ {{1, 12}, {2, 2}, {3, 3}, {4, 20}, {5, 5}, {6, 27}, {7, 7}, {8, 8}, {9, 11}, \ {10, 19}, {11, 26}, {12, 33}, {13, 13}, {14, 14}, {15, 15}, {16, 16}, {17, 17}, {18, 10}, {19, \ 25}, {20, 36}, {21, 21}, {22, 22}, {23, 23}, {24, 9}, {25, 18}, {26, 24}, {27, 38}, {28, 28}, {29, \ 29}, {30, 30}, {31, 31}, {32, 32}, {33, 41}, {34, 34}, {35, 35}, {36, 44}, {37, 37}, {38, 46}, {39, \ 39}, {40, 40}, {41, 1}, {42, 42}, {43, 43}, {44, 4}, {45, 45}, {46, 6}, {47, 47}, {48, 48}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["gen3 = Perm @@ Transpose[l][[2]]", "Input"], Cell[OutputFormData["\<\ Perm[12, 2, 3, 20, 5, 27, 7, 8, 11, 19, 26, 33, 13, 14, 15, 16, 17, 10, 25, \ 36, 21, 22, 23, 9, 18, 24, 38, 28, 29, 30, 31, 32, 41, 34, 35, 44, 37, 46, 39, 40, 1, 42, 43, 4, 45, \ 6, 47, 48]\ \>", "\<\ ( 12 2 3 20 5 27 7 8 11 19 26 33 13 14 15 16 17 10 25 36 21 22 23 9 18 24 38 \ 28 29 30 31 32 41 34 35 44\\ 37 46 39 40 1 42 43 4 45 6 47 48 )\ \>"], "Output"] }, Open ]], Cell["Generator gen4: Rotation of A 41 => 43 => 48 => 46 => 41", "Text"], Cell[CellGroupData[{ Cell["\<\ l = {{1, 24}, {2, 18}, {3, 9}, {4, 4}, {5, 5}, {6, 6}, {7, 7}, {8, 8}, {9, \ 38}, {10, 10}, {11, 11}, {12, 12}, {13, 13}, {14, 14}, {15, 15}, {16, 16}, {17, 1}, {18, 39}, {19, 19}, {20, 20}, {21, \ 21}, {22, 22}, {23, 2}, {24, 40}, {25, 25}, {26, 26}, {27, 27}, {28, 28}, {29, 29}, {30, 30}, {31, 31}, {32, \ 3}, {33, 33}, {34, 34}, {35, 35}, {36, 36}, {37, 37}, {38, 32}, {39, 23}, {40, 17}, {41, 43}, {42, 45}, {43, \ 48}, {44, 42}, {45, 47}, {46, 41}, {47, 44}, {48, 46}}\ \>", "Input"], Cell[OutputFormData["\<\ {{1, 24}, {2, 18}, {3, 9}, {4, 4}, {5, 5}, {6, 6}, {7, 7}, {8, 8}, {9, 38}, \ {10, 10}, {11, 11}, {12, 12}, {13, 13}, {14, 14}, {15, 15}, {16, 16}, {17, 1}, {18, 39}, {19, 19}, {20, \ 20}, {21, 21}, {22, 22}, {23, 2}, {24, 40}, {25, 25}, {26, 26}, {27, 27}, {28, 28}, {29, 29}, {30, \ 30}, {31, 31}, {32, 3}, {33, 33}, {34, 34}, {35, 35}, {36, 36}, {37, 37}, {38, 32}, {39, 23}, {40, \ 17}, {41, 43}, {42, 45}, {43, 48}, {44, 42}, {45, 47}, {46, 41}, {47, 44}, {48, 46}}\ \>", "\<\ {{1, 24}, {2, 18}, {3, 9}, {4, 4}, {5, 5}, {6, 6}, {7, 7}, {8, 8}, {9, 38}, \ {10, 10}, {11, 11}, {12, 12}, {13, 13}, {14, 14}, {15, 15}, {16, 16}, {17, 1}, {18, 39}, {19, 19}, {20, \ 20}, {21, 21}, {22, 22}, {23, 2}, {24, 40}, {25, 25}, {26, 26}, {27, 27}, {28, 28}, {29, 29}, {30, \ 30}, {31, 31}, {32, 3}, {33, 33}, {34, 34}, {35, 35}, {36, 36}, {37, 37}, {38, 32}, {39, 23}, {40, \ 17}, {41, 43}, {42, 45}, {43, 48}, {44, 42}, {45, 47}, {46, 41}, {47, 44}, {48, 46}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["gen4 = Perm @@ Transpose[l][[2]]", "Input"], Cell[OutputFormData["\<\ Perm[24, 18, 9, 4, 5, 6, 7, 8, 38, 10, 11, 12, 13, 14, 15, 16, 1, 39, 19, 20, \ 21, 22, 2, 40, 25, 26, 27, 28, 29, 30, 31, 3, 33, 34, 35, 36, 37, 32, 23, 17, 43, 45, 48, 42, 47, 41, \ 44, 46]\ \>", "\<\ ( 24 18 9 4 5 6 7 8 38 10 11 12 13 14 15 16 1 39 19 20 21 22 2 40 25 26 27 28 \ 29 30 31 3 33 34 35 36 37\\ 32 23 17 43 45 48 42 47 41 44 46 )\ \>"], "Output"] }, Open ]], Cell["Generator gen5: Rotation of D 17 => 32 => 30 => 15 => 17", "Text"], Cell[CellGroupData[{ Cell["\<\ l = {{1, 1}, {2, 2}, {3, 43}, {4, 4}, {5, 45}, {6, 6}, {7, 7}, {8, 48}, {9, \ 9}, {10, 10}, {11, 11}, {12, 12}, {13, 13}, {14, 3}, {15, 17}, {16, 23}, {17, 32}, {18, 18}, {19, 19}, {20, 20}, {21, \ 5}, {22, 16}, {23, 31}, {24, 24}, {25, 25}, {26, 26}, {27, 27}, {28, 28}, {29, 8}, {30, 15}, {31, 22}, {32, \ 30}, {33, 33}, {34, 34}, {35, 14}, {36, 36}, {37, 21}, {38, 38}, {39, 39}, {40, 29}, {41, 41}, {42, 42}, {43, \ 35}, {44, 44}, {45, 37}, {46, 46}, {47, 47}, {48, 40}}\ \>", "Input"], Cell[OutputFormData["\<\ {{1, 1}, {2, 2}, {3, 43}, {4, 4}, {5, 45}, {6, 6}, {7, 7}, {8, 48}, {9, 9}, \ {10, 10}, {11, 11}, {12, 12}, {13, 13}, {14, 3}, {15, 17}, {16, 23}, {17, 32}, {18, 18}, {19, 19}, {20, \ 20}, {21, 5}, {22, 16}, {23, 31}, {24, 24}, {25, 25}, {26, 26}, {27, 27}, {28, 28}, {29, 8}, {30, \ 15}, {31, 22}, {32, 30}, {33, 33}, {34, 34}, {35, 14}, {36, 36}, {37, 21}, {38, 38}, {39, 39}, {40, \ 29}, {41, 41}, {42, 42}, {43, 35}, {44, 44}, {45, 37}, {46, 46}, {47, 47}, {48, 40}}\ \>", "\<\ {{1, 1}, {2, 2}, {3, 43}, {4, 4}, {5, 45}, {6, 6}, {7, 7}, {8, 48}, {9, 9}, \ {10, 10}, {11, 11}, {12, 12}, {13, 13}, {14, 3}, {15, 17}, {16, 23}, {17, 32}, {18, 18}, {19, 19}, {20, \ 20}, {21, 5}, {22, 16}, {23, 31}, {24, 24}, {25, 25}, {26, 26}, {27, 27}, {28, 28}, {29, 8}, {30, \ 15}, {31, 22}, {32, 30}, {33, 33}, {34, 34}, {35, 14}, {36, 36}, {37, 21}, {38, 38}, {39, 39}, {40, \ 29}, {41, 41}, {42, 42}, {43, 35}, {44, 44}, {45, 37}, {46, 46}, {47, 47}, {48, 40}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["gen5 = Perm @@ Transpose[l][[2]]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 43, 4, 45, 6, 7, 48, 9, 10, 11, 12, 13, 3, 17, 23, 32, 18, 19, 20, \ 5, 16, 31, 24, 25, 26, 27, 28, 8, 15, 22, 30, 33, 34, 14, 36, 21, 38, 39, 29, 41, 42, 35, 44, 37, 46, \ 47, 40]\ \>", "\<\ ( 1 2 43 4 45 6 7 48 9 10 11 12 13 3 17 23 32 18 19 20 5 16 31 24 25 26 27 28 \ 8 15 22 30 33 34 14 36 21\\ 38 39 29 41 42 35 44 37 46 47 40 )\ \>"], "Output"] }, Open ]], Cell["Generator gen6: Rotation of F 33 => 35 => 40 => 38 => 33", "Text"], Cell[CellGroupData[{ Cell["\<\ l = {{1, 1}, {2, 2}, {3, 3}, {4, 4}, {5, 5}, {6, 6}, {7, 7}, {8, 8}, {9, 9}, \ {10, 10}, {11, 11}, {12, 12}, {13, 13}, {14, 14}, {15, 15}, {16, 16}, {17, 17}, {18, 18}, {19, 19}, {20, 20}, {21, \ 21}, {22, 22}, {23, 23}, {24, 27}, {25, 28}, {26, 29}, {27, 30}, {28, 31}, {29, 32}, {30, 43}, {31, 42}, {32, \ 41}, {33, 35}, {34, 37}, {35, 40}, {36, 34}, {37, 39}, {38, 33}, {39, 36}, {40, 38}, {41, 26}, {42, 25}, {43, \ 24}, {44, 44}, {45, 45}, {46, 46}, {47, 47}, {48, 48}}\ \>", "Input"], Cell[OutputFormData["\<\ {{1, 1}, {2, 2}, {3, 3}, {4, 4}, {5, 5}, {6, 6}, {7, 7}, {8, 8}, {9, 9}, {10, \ 10}, {11, 11}, {12, 12}, {13, 13}, {14, 14}, {15, 15}, {16, 16}, {17, 17}, {18, 18}, {19, 19}, {20, \ 20}, {21, 21}, {22, 22}, {23, 23}, {24, 27}, {25, 28}, {26, 29}, {27, 30}, {28, 31}, {29, 32}, {30, \ 43}, {31, 42}, {32, 41}, {33, 35}, {34, 37}, {35, 40}, {36, 34}, {37, 39}, {38, 33}, {39, 36}, {40, \ 38}, {41, 26}, {42, 25}, {43, 24}, {44, 44}, {45, 45}, {46, 46}, {47, 47}, {48, 48}}\ \>", "\<\ {{1, 1}, {2, 2}, {3, 3}, {4, 4}, {5, 5}, {6, 6}, {7, 7}, {8, 8}, {9, 9}, {10, \ 10}, {11, 11}, {12, 12}, {13, 13}, {14, 14}, {15, 15}, {16, 16}, {17, 17}, {18, 18}, {19, 19}, {20, \ 20}, {21, 21}, {22, 22}, {23, 23}, {24, 27}, {25, 28}, {26, 29}, {27, 30}, {28, 31}, {29, 32}, {30, \ 43}, {31, 42}, {32, 41}, {33, 35}, {34, 37}, {35, 40}, {36, 34}, {37, 39}, {38, 33}, {39, 36}, {40, \ 38}, {41, 26}, {42, 25}, {43, 24}, {44, 44}, {45, 45}, {46, 46}, {47, 47}, {48, 48}}\ \>"], "Output"] }, Open ]], Cell[CellGroupData[{ Cell["gen6 = Perm @@ Transpose[l][[2]]", "Input"], Cell[OutputFormData["\<\ Perm[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, \ 21, 22, 23, 27, 28, 29, 30, 31, 32, 43, 42, 41, 35, 37, 40, 34, 39, 33, 36, 38, 26, 25, 24, 44, 45, 46, \ 47, 48]\ \>", "\<\ ( 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 27 28 29 30 31 \ 32 43 42 41 35 37 40 34 39\\ 33 36 38 26 25 24 44 45 46 47 48 )\ \>"], "Output"] }, Open ]], Cell["List of all generators", "Text"], Cell[CellGroupData[{ Cell["gens = HoldList @@ {gen1, gen2, gen3, gen4, gen5, gen6}", "Input"], Cell[OutputFormData["\<\ HoldList[Perm[3, 5, 8, 2, 7, 1, 4, 6, 48, 47, 46, 9, 10, 11, 12, 13, 14, 18, \ 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, \ 44, 45, 17, 16, 15], Perm[1, 2, 3, 4, 5, 15, 22, 30, 9, 10, 8, 14, 21, 29, 35, 16, 17, 18, 7, \ 13, 28, 34, 23, 24, 25, 6, 12, 20, 27, 33, 31, 32, 11, 19, 26, 36, 37, 38, 39, 40, 41, 42, 43, 44, \ 45, 46, 47, 48], Perm[12, 2, 3, 20, 5, 27, 7, 8, 11, 19, 26, 33, 13, 14, 15, 16, 17, 10, 25, \ 36, 21, 22, 23, 9, 18, 24, 38, 28, 29, 30, 31, 32, 41, 34, 35, 44, 37, 46, 39, 40, 1, 42, 43, 4, 45, \ 6, 47, 48], Perm[24, 18, 9, 4, 5, 6, 7, 8, 38, 10, 11, 12, 13, 14, 15, 16, 1, 39, 19, \ 20, 21, 22, 2, 40, 25, 26, 27, 28, 29, 30, 31, 3, 33, 34, 35, 36, 37, 32, 23, 17, 43, 45, 48, 42, 47, \ 41, 44, 46], Perm[1, 2, 43, 4, 45, 6, 7, 48, 9, 10, 11, 12, 13, 3, 17, 23, 32, 18, 19, \ 20, 5, 16, 31, 24, 25, 26, 27, 28, 8, 15, 22, 30, 33, 34, 14, 36, 21, 38, 39, 29, 41, 42, 35, 44, 37, \ 46, 47, 40], Perm[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, \ 21, 22, 23, 27, 28, 29, 30, 31, 32, 43, 42, 41, 35, 37, 40, 34, 39, 33, 36, 38, 26, 25, 24, 44, 45, \ 46, 47, 48]]\ \>", "\<\ {( 3 5 8 2 7 1 4 6 48 47 46 9 10 11 12 13 14 18 19 20 21 22 23 24 25 26 27 28 \ 29 30 31 32 33 34 35 36 37\\ 38 39 40 41 42 43 44 45 17 16 15 ), ( 1 2 3 4 5 15 22 30 9 10 8 14 21 29 \ 35 16 17 18 7 13 28 34 23 24\\ 25 6 12 20 27 33 31 32 11 19 26 36 37 38 39 40 41 42 43 44 45 46 47 48 ), ( 12 2 3 20 5 27 7 8 11 19 26 33 13 14 15 16 17 10 25 36 21 22 23 9 18 24 \ 38 28 29 30 31 32 41 34 35 44\\ 37 46 39 40 1 42 43 4 45 6 47 48 ), ( 24 18 9 4 5 6 7 8 38 10 11 12 13 14 \ 15 16 1 39 19 20 21 22 2 40\\ 25 26 27 28 29 30 31 3 33 34 35 36 37 32 23 17 43 45 48 42 47 41 44 46 ), ( 1 2 43 4 45 6 7 48 9 10 11 12 13 3 17 23 32 18 19 20 5 16 31 24 25 26 27 \ 28 8 15 22 30 33 34 14 36 21\\ 38 39 29 41 42 35 44 37 46 47 40 ), ( 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 \ 16 17 18 19 20 21 22 23 27\\ 28 29 30 31 32 43 42 41 35 37 40 34 39 33 36 38 26 25 24 44 45 46 47 48 )}\ \ \>"], "Output"] }, Open ]], Cell["\<\ Calculation of all orbits, a Schreier vector and a backward pointer list\ \>", "Text"], Cell[CellGroupData[{ Cell["orbits = AllOrbits[gens]", "Input"], Cell[OutputFormData["\<\ HoldList[HoldList[HoldList[1, 3, 12, 24, 8, 9, 43, 14, 33, 40, 27, 6, 30, 48, \ 11, 38, 35, 29, 41, 17, 15, 46, 26, 32], HoldList[2, 5, 18, 7, 45, 10, 39, 4, 22, 47, 37, 19, 23, \ 36, 20, 34, 16, 44, 21, 25, 31, 13, 42, 28]], HoldList[0, 0, Perm[3, 5, 8, 2, 7, 1, 4, 6, 48, 47, \ 46, 9, 10, 11, 12, 13, 14, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, \ 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 17, 16, 15], Perm[3, 5, 8, 2, 7, 1, 4, 6, 48, 47, 46, 9, \ 10, 11, 12, 13, 14, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, \ 37, 38, 39, 40, 41, 42, 43, 44, 45, 17, 16, 15], Perm[3, 5, 8, 2, 7, 1, 4, 6, 48, 47, 46, 9, 10, 11, \ 12, 13, 14, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, \ 39, 40, 41, 42, 43, 44, 45, 17, 16, 15], Perm[3, 5, 8, 2, 7, 1, 4, 6, 48, 47, 46, 9, 10, 11, 12, 13, \ 14, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, \ 41, 42, 43, 44, 45, 17, 16, 15], Perm[3, 5, 8, 2, 7, 1, 4, 6, 48, 47, 46, 9, 10, 11, 12, 13, 14, 18, \ 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, \ 43, 44, 45, 17, 16, 15], Perm[3, 5, 8, 2, 7, 1, 4, 6, 48, 47, 46, 9, 10, 11, 12, 13, 14, 18, 19, \ 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, \ 45, 17, 16, 15], Perm[24, 18, 9, 4, 5, 6, 7, 8, 38, 10, 11, 12, 13, 14, 15, 16, 1, 39, 19, \ 20, 21, 22, 2, 40, 25, 26, 27, 28, 29, 30, 31, 3, 33, 34, 35, 36, 37, 32, 23, 17, 43, 45, 48, 42, \ 47, 41, 44, 46], Perm[12, 2, 3, 20, 5, 27, 7, 8, 11, 19, 26, 33, 13, 14, 15, 16, 17, 10, \ 25, 36, 21, 22, 23, 9, 18, 24, 38, 28, 29, 30, 31, 32, 41, 34, 35, 44, 37, 46, 39, 40, 1, 42, 43, 4, \ 45, 6, 47, 48], Perm[12, 2, 3, 20, 5, 27, 7, 8, 11, 19, 26, 33, 13, 14, 15, 16, 17, 10, \ 25, 36, 21, 22, 23, 9, 18, 24, 38, 28, 29, 30, 31, 32, 41, 34, 35, 44, 37, 46, 39, 40, 1, 42, 43, 4, \ 45, 6, 47, 48], Perm[12, 2, 3, 20, 5, 27, 7, 8, 11, 19, 26, 33, 13, 14, 15, 16, 17, 10, \ 25, 36, 21, 22, 23, 9, 18, 24, 38, 28, 29, 30, 31, 32, 41, 34, 35, 44, 37, 46, 39, 40, 1, 42, 43, 4, \ 45, 6, 47, 48], Perm[1, 2, 3, 4, 5, 15, 22, 30, 9, 10, 8, 14, 21, 29, 35, 16, 17, 18, 7, \ 13, 28, 34, 23, 24, 25, 6, 12, 20, 27, 33, 31, 32, 11, 19, 26, 36, 37, 38, 39, 40, 41, 42, 43, 44, \ 45, 46, 47, 48], Perm[1, 2, 3, 4, 5, 15, 22, 30, 9, 10, 8, 14, 21, 29, 35, 16, 17, 18, 7, \ 13, 28, 34, 23, 24, 25, 6, 12, 20, 27, 33, 31, 32, 11, 19, 26, 36, 37, 38, 39, 40, 41, 42, 43, 44, \ 45, 46, 47, 48], Perm[1, 2, 3, 4, 5, 15, 22, 30, 9, 10, 8, 14, 21, 29, 35, 16, 17, 18, 7, \ 13, 28, 34, 23, 24, 25, 6, 12, 20, 27, 33, 31, 32, 11, 19, 26, 36, 37, 38, 39, 40, 41, 42, 43, 44, \ 45, 46, 47, 48], Perm[1, 2, 43, 4, 45, 6, 7, 48, 9, 10, 11, 12, 13, 3, 17, 23, 32, 18, 19, \ 20, 5, 16, 31, 24, 25, 26, 27, 28, 8, 15, 22, 30, 33, 34, 14, 36, 21, 38, 39, 29, 41, 42, 35, 44, \ 37, 46, 47, 40], Perm[24, 18, 9, 4, 5, 6, 7, 8, 38, 10, 11, 12, 13, 14, 15, 16, 1, 39, 19, \ 20, 21, 22, 2, 40, 25, 26, 27, 28, 29, 30, 31, 3, 33, 34, 35, 36, 37, 32, 23, 17, 43, 45, 48, 42, \ 47, 41, 44, 46], Perm[24, 18, 9, 4, 5, 6, 7, 8, 38, 10, 11, 12, 13, 14, 15, 16, 1, 39, 19, \ 20, 21, 22, 2, 40, 25, 26, 27, 28, 29, 30, 31, 3, 33, 34, 35, 36, 37, 32, 23, 17, 43, 45, 48, 42, \ 47, 41, 44, 46], Perm[12, 2, 3, 20, 5, 27, 7, 8, 11, 19, 26, 33, 13, 14, 15, 16, 17, 10, \ 25, 36, 21, 22, 23, 9, 18, 24, 38, 28, 29, 30, 31, 32, 41, 34, 35, 44, 37, 46, 39, 40, 1, 42, 43, 4, \ 45, 6, 47, 48], Perm[12, 2, 3, 20, 5, 27, 7, 8, 11, 19, 26, 33, 13, 14, 15, 16, 17, 10, \ 25, 36, 21, 22, 23, 9, 18, 24, 38, 28, 29, 30, 31, 32, 41, 34, 35, 44, 37, 46, 39, 40, 1, 42, 43, 4, \ 45, 6, 47, 48], Perm[1, 2, 43, 4, 45, 6, 7, 48, 9, 10, 11, 12, 13, 3, 17, 23, 32, 18, 19, \ 20, 5, 16, 31, 24, 25, 26, 27, 28, 8, 15, 22, 30, 33, 34, 14, 36, 21, 38, 39, 29, 41, 42, 35, 44, \ 37, 46, 47, 40], Perm[1, 2, 3, 4, 5, 15, 22, 30, 9, 10, 8, 14, 21, 29, 35, 16, 17, 18, 7, \ 13, 28, 34, 23, 24, 25, 6, 12, 20, 27, 33, 31, 32, 11, 19, 26, 36, 37, 38, 39, 40, 41, 42, 43, 44, \ 45, 46, 47, 48], Perm[24, 18, 9, 4, 5, 6, 7, 8, 38, 10, 11, 12, 13, 14, 15, 16, 1, 39, 19, \ 20, 21, 22, 2, 40, 25, 26, 27, 28, 29, 30, 31, 3, 33, 34, 35, 36, 37, 32, 23, 17, 43, 45, 48, 42, \ 47, 41, 44, 46], Perm[24, 18, 9, 4, 5, 6, 7, 8, 38, 10, 11, 12, 13, 14, 15, 16, 1, 39, 19, \ 20, 21, 22, 2, 40, 25, 26, 27, 28, 29, 30, 31, 3, 33, 34, 35, 36, 37, 32, 23, 17, 43, 45, 48, 42, \ 47, 41, 44, 46], Perm[12, 2, 3, 20, 5, 27, 7, 8, 11, 19, 26, 33, 13, 14, 15, 16, 17, 10, \ 25, 36, 21, 22, 23, 9, 18, 24, 38, 28, 29, 30, 31, 32, 41, 34, 35, 44, 37, 46, 39, 40, 1, 42, 43, 4, \ 45, 6, 47, 48], Perm[12, 2, 3, 20, 5, 27, 7, 8, 11, 19, 26, 33, 13, 14, 15, 16, 17, 10, \ 25, 36, 21, 22, 23, 9, 18, 24, 38, 28, 29, 30, 31, 32, 41, 34, 35, 44, 37, 46, 39, 40, 1, 42, 43, 4, \ 45, 6, 47, 48], Perm[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, \ 20, 21, 22, 23, 27, 28, 29, 30, 31, 32, 43, 42, 41, 35, 37, 40, 34, 39, 33, 36, 38, 26, 25, 24, 44, \ 45, 46, 47, 48], Perm[1, 2, 3, 4, 5, 15, 22, 30, 9, 10, 8, 14, 21, 29, 35, 16, 17, 18, 7, \ 13, 28, 34, 23, 24, 25, 6, 12, 20, 27, 33, 31, 32, 11, 19, 26, 36, 37, 38, 39, 40, 41, 42, 43, 44, \ 45, 46, 47, 48], Perm[1, 2, 3, 4, 5, 15, 22, 30, 9, 10, 8, 14, 21, 29, 35, 16, 17, 18, 7, \ 13, 28, 34, 23, 24, 25, 6, 12, 20, 27, 33, 31, 32, 11, 19, 26, 36, 37, 38, 39, 40, 41, 42, 43, 44, \ 45, 46, 47, 48], Perm[1, 2, 3, 4, 5, 15, 22, 30, 9, 10, 8, 14, 21, 29, 35, 16, 17, 18, 7, \ 13, 28, 34, 23, 24, 25, 6, 12, 20, 27, 33, 31, 32, 11, 19, 26, 36, 37, 38, 39, 40, 41, 42, 43, 44, \ 45, 46, 47, 48], Perm[1, 2, 43, 4, 45, 6, 7, 48, 9, 10, 11, 12, 13, 3, 17, 23, 32, 18, 19, \ 20, 5, 16, 31, 24, 25, 26, 27, 28, 8, 15, 22, 30, 33, 34, 14, 36, 21, 38, 39, 29, 41, 42, 35, 44, \ 37, 46, 47, 40], Perm[24, 18, 9, 4, 5, 6, 7, 8, 38, 10, 11, 12, 13, 14, 15, 16, 1, 39, 19, \ 20, 21, 22, 2, 40, 25, 26, 27, 28, 29, 30, 31, 3, 33, 34, 35, 36, 37, 32, 23, 17, 43, 45, 48, 42, \ 47, 41, 44, 46], Perm[12, 2, 3, 20, 5, 27, 7, 8, 11, 19, 26, 33, 13, 14, 15, 16, 17, 10, \ 25, 36, 21, 22, 23, 9, 18, 24, 38, 28, 29, 30, 31, 32, 41, 34, 35, 44, 37, 46, 39, 40, 1, 42, 43, 4, \ 45, 6, 47, 48], Perm[1, 2, 3, 4, 5, 15, 22, 30, 9, 10, 8, 14, 21, 29, 35, 16, 17, 18, 7, \ 13, 28, 34, 23, 24, 25, 6, 12, 20, 27, 33, 31, 32, 11, 19, 26, 36, 37, 38, 39, 40, 41, 42, 43, 44, \ 45, 46, 47, 48], Perm[1, 2, 43, 4, 45, 6, 7, 48, 9, 10, 11, 12, 13, 3, 17, 23, 32, 18, 19, \ 20, 5, 16, 31, 24, 25, 26, 27, 28, 8, 15, 22, 30, 33, 34, 14, 36, 21, 38, 39, 29, 41, 42, 35, 44, \ 37, 46, 47, 40], Perm[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, \ 20, 21, 22, 23, 27, 28, 29, 30, 31, 32, 43, 42, 41, 35, 37, 40, 34, 39, 33, 36, 38, 26, 25, 24, 44, \ 45, 46, 47, 48], Perm[1, 2, 43, 4, 45, 6, 7, 48, 9, 10, 11, 12, 13, 3, 17, 23, 32, 18, 19, \ 20, 5, 16, 31, 24, 25, 26, 27, 28, 8, 15, 22, 30, 33, 34, 14, 36, 21, 38, 39, 29, 41, 42, 35, 44, \ 37, 46, 47, 40], Perm[24, 18, 9, 4, 5, 6, 7, 8, 38, 10, 11, 12, 13, 14, 15, 16, 1, 39, 19, \ 20, 21, 22, 2, 40, 25, 26, 27, 28, 29, 30, 31, 3, 33, 34, 35, 36, 37, 32, 23, 17, 43, 45, 48, 42, \ 47, 41, 44, 46], Perm[24, 18, 9, 4, 5, 6, 7, 8, 38, 10, 11, 12, 13, 14, 15, 16, 1, 39, 19, \ 20, 21, 22, 2, 40, 25, 26, 27, 28, 29, 30, 31, 3, 33, 34, 35, 36, 37, 32, 23, 17, 43, 45, 48, 42, \ 47, 41, 44, 46], Perm[24, 18, 9, 4, 5, 6, 7, 8, 38, 10, 11, 12, 13, 14, 15, 16, 1, 39, 19, \ 20, 21, 22, 2, 40, 25, 26, 27, 28, 29, 30, 31, 3, 33, 34, 35, 36, 37, 32, 23, 17, 43, 45, 48, 42, \ 47, 41, 44, 46], Perm[12, 2, 3, 20, 5, 27, 7, 8, 11, 19, 26, 33, 13, 14, 15, 16, 17, 10, \ 25, 36, 21, 22, 23, 9, 18, 24, 38, 28, 29, 30, 31, 32, 41, 34, 35, 44, 37, 46, 39, 40, 1, 42, 43, 4, \ 45, 6, 47, 48], Perm[24, 18, 9, 4, 5, 6, 7, 8, 38, 10, 11, 12, 13, 14, 15, 16, 1, 39, 19, \ 20, 21, 22, 2, 40, 25, 26, 27, 28, 29, 30, 31, 3, 33, 34, 35, 36, 37, 32, 23, 17, 43, 45, 48, 42, \ 47, 41, 44, 46], Perm[1, 2, 43, 4, 45, 6, 7, 48, 9, 10, 11, 12, 13, 3, 17, 23, 32, 18, 19, \ 20, 5, 16, 31, 24, 25, 26, 27, 28, 8, 15, 22, 30, 33, 34, 14, 36, 21, 38, 39, 29, 41, 42, 35, 44, \ 37, 46, 47, 40], Perm[24, 18, 9, 4, 5, 6, 7, 8, 38, 10, 11, 12, 13, 14, 15, 16, 1, 39, 19, \ 20, 21, 22, 2, 40, 25, 26, 27, 28, 29, 30, 31, 3, 33, 34, 35, 36, 37, 32, 23, 17, 43, 45, 48, 42, \ 47, 41, 44, 46], Perm[1, 2, 43, 4, 45, 6, 7, 48, 9, 10, 11, 12, 13, 3, 17, 23, 32, 18, 19, \ 20, 5, 16, 31, 24, 25, 26, 27, 28, 8, 15, 22, 30, 33, 34, 14, 36, 21, 38, 39, 29, 41, 42, 35, 44, \ 37, 46, 47, 40], Perm[24, 18, 9, 4, 5, 6, 7, 8, 38, 10, 11, 12, 13, 14, 15, 16, 1, 39, 19, \ 20, 21, 22, 2, 40, 25, 26, 27, 28, 29, 30, 31, 3, 33, 34, 35, 36, 37, 32, 23, 17, 43, 45, 48, 42, \ 47, 41, 44, 46], Perm[24, 18, 9, 4, 5, 6, 7, 8, 38, 10, 11, 12, 13, 14, 15, 16, 1, 39, 19, \ 20, 21, 22, 2, 40, 25, 26, 27, 28, 29, 30, 31, 3, 33, 34, 35, 36, 37, 32, 23, 17, 43, 45, 48, 42, \ 47, 41, 44, 46], Perm[1, 2, 43, 4, 45, 6, 7, 48, 9, 10, 11, 12, 13, 3, 17, 23, 32, 18, 19, \ 20, 5, 16, 31, 24, 25, 26, 27, 28, 8, 15, 22, 30, 33, 34, 14, 36, 21, 38, 39, 29, 41, 42, 35, 44, \ 37, 46, 47, 40]], HoldList[0, 0, 1, 7, 2, 8, 5, 3, 3, 18, 9, 1, 20, 12, 6, 22, 40, 2, 10, 4, \ 37, 7, 39, 1, 19, 11, 24, 21, 14, 8, 23, 38, 12, 22, 43, 39, 45, 9, 18, 24, 33, 44, 3, 47, 5, 48, \ 45, 8]]\ \>", "\<\ {{{1, 3, 12, 24, 8, 9, 43, 14, 33, 40, 27, 6, 30, 48, 11, 38, 35, 29, 41, 17, \ 15, 46, 26, 32}, {2, 5, 18, 7, 45, 10, 39, 4, 22, 47, 37, 19, 23, 36, 20, 34, 16, 44, 21, \ 25, 31, 13, 42, 28}}, {0, 0, ( 3 5 8 2 7 1 4 6 48 47 46 9 10 11 12 13 14 18 19 20 21 22 23 24 25 \ 26 27 28 29 30 31 32 33 34\\ 35 36 37 38 39 40 41 42 43 44 45 17 16 15 ), ( 3 5 8 2 7 1 4 6 48 47 46 9 10 11 12 13 14 18 19 20 21 22 23 24 25 26 27 \ 28 29 30 31 32 33 34 35 36\\ 37 38 39 40 41 42 43 44 45 17 16 15 ), ( 3 5 8 2 7 1 4 6 48 47 46 9 10 11 12 13 14 18 19 20 21 22 23 24 25 26 27 \ 28 29 30 31 32 33 34 35 36\\ 37 38 39 40 41 42 43 44 45 17 16 15 ), ( 3 5 8 2 7 1 4 6 48 47 46 9 10 11 12 13 14 18 19 20 21 22 23 24 25 26 27 \ 28 29 30 31 32 33 34 35 36\\ 37 38 39 40 41 42 43 44 45 17 16 15 ), ( 3 5 8 2 7 1 4 6 48 47 46 9 10 11 12 13 14 18 19 20 21 22 23 24 25 26 27 \ 28 29 30 31 32 33 34 35 36\\ 37 38 39 40 41 42 43 44 45 17 16 15 ), ( 3 5 8 2 7 1 4 6 48 47 46 9 10 11 12 13 14 18 19 20 21 22 23 24 25 26 27 \ 28 29 30 31 32 33 34 35 36\\ 37 38 39 40 41 42 43 44 45 17 16 15 ), ( 24 18 9 4 5 6 7 8 38 10 11 12 13 14 15 16 1 39 19 20 21 22 2 40 25 26 27 \ 28 29 30 31 3 33 34 35 36\\ 37 32 23 17 43 45 48 42 47 41 44 46 ), ( 12 2 3 20 5 27 7 8 11 19 26 33 13 14 15 16 17 10 25 36 21 22 23 9 18 24 \ 38 28 29 30 31 32 41 34 35\\ 44 37 46 39 40 1 42 43 4 45 6 47 48 ), ( 12 2 3 20 5 27 7 8 11 19 26 33 13 14 15 16 17 10 25 36 21 22 23 9 18 24 \ 38 28 29 30 31 32 41 34 35\\ 44 37 46 39 40 1 42 43 4 45 6 47 48 ), ( 12 2 3 20 5 27 7 8 11 19 26 33 13 14 15 16 17 10 25 36 21 22 23 9 18 24 \ 38 28 29 30 31 32 41 34 35\\ 44 37 46 39 40 1 42 43 4 45 6 47 48 ), ( 1 2 3 4 5 15 22 30 9 10 8 14 21 29 35 16 17 18 7 13 28 34 23 24 25 6 12 \ 20 27 33 31 32 11 19 26 36\\ 37 38 39 40 41 42 43 44 45 46 47 48 ), ( 1 2 3 4 5 15 22 30 9 10 8 14 21 29 35 16 17 18 7 13 28 34 23 24 25 6 12 \ 20 27 33 31 32 11 19 26 36\\ 37 38 39 40 41 42 43 44 45 46 47 48 ), ( 1 2 3 4 5 15 22 30 9 10 8 14 21 29 35 16 17 18 7 13 28 34 23 24 25 6 12 \ 20 27 33 31 32 11 19 26 36\\ 37 38 39 40 41 42 43 44 45 46 47 48 ), ( 1 2 43 4 45 6 7 48 9 10 11 12 13 3 17 23 32 18 19 20 5 16 31 24 25 26 27 \ 28 8 15 22 30 33 34 14 36\\ 21 38 39 29 41 42 35 44 37 46 47 40 ), ( 24 18 9 4 5 6 7 8 38 10 11 12 13 14 15 16 1 39 19 20 21 22 2 40 25 26 27 \ 28 29 30 31 3 33 34 35 36\\ 37 32 23 17 43 45 48 42 47 41 44 46 ), ( 24 18 9 4 5 6 7 8 38 10 11 12 13 14 15 16 1 39 19 20 21 22 2 40 25 26 27 \ 28 29 30 31 3 33 34 35 36\\ 37 32 23 17 43 45 48 42 47 41 44 46 ), ( 12 2 3 20 5 27 7 8 11 19 26 33 13 14 15 16 17 10 25 36 21 22 23 9 18 24 \ 38 28 29 30 31 32 41 34 35\\ 44 37 46 39 40 1 42 43 4 45 6 47 48 ), ( 12 2 3 20 5 27 7 8 11 19 26 33 13 14 15 16 17 10 25 36 21 22 23 9 18 24 \ 38 28 29 30 31 32 41 34 35\\ 44 37 46 39 40 1 42 43 4 45 6 47 48 ), ( 1 2 43 4 45 6 7 48 9 10 11 12 13 3 17 23 32 18 19 20 5 16 31 24 25 26 27 \ 28 8 15 22 30 33 34 14 36\\ 21 38 39 29 41 42 35 44 37 46 47 40 ), ( 1 2 3 4 5 15 22 30 9 10 8 14 21 29 35 16 17 18 7 13 28 34 23 24 25 6 12 \ 20 27 33 31 32 11 19 26 36\\ 37 38 39 40 41 42 43 44 45 46 47 48 ), ( 24 18 9 4 5 6 7 8 38 10 11 12 13 14 15 16 1 39 19 20 21 22 2 40 25 26 27 \ 28 29 30 31 3 33 34 35 36\\ 37 32 23 17 43 45 48 42 47 41 44 46 ), ( 24 18 9 4 5 6 7 8 38 10 11 12 13 14 15 16 1 39 19 20 21 22 2 40 25 26 27 \ 28 29 30 31 3 33 34 35 36\\ 37 32 23 17 43 45 48 42 47 41 44 46 ), ( 12 2 3 20 5 27 7 8 11 19 26 33 13 14 15 16 17 10 25 36 21 22 23 9 18 24 \ 38 28 29 30 31 32 41 34 35\\ 44 37 46 39 40 1 42 43 4 45 6 47 48 ), ( 12 2 3 20 5 27 7 8 11 19 26 33 13 14 15 16 17 10 25 36 21 22 23 9 18 24 \ 38 28 29 30 31 32 41 34 35\\ 44 37 46 39 40 1 42 43 4 45 6 47 48 ), ( 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 27 28 29 30 \ 31 32 43 42 41 35 37 40 34\\ 39 33 36 38 26 25 24 44 45 46 47 48 ), ( 1 2 3 4 5 15 22 30 9 10 8 14 21 29 35 16 17 18 7 13 28 34 23 24 25 6 12 \ 20 27 33 31 32 11 19 26 36\\ 37 38 39 40 41 42 43 44 45 46 47 48 ), ( 1 2 3 4 5 15 22 30 9 10 8 14 21 29 35 16 17 18 7 13 28 34 23 24 25 6 12 \ 20 27 33 31 32 11 19 26 36\\ 37 38 39 40 41 42 43 44 45 46 47 48 ), ( 1 2 3 4 5 15 22 30 9 10 8 14 21 29 35 16 17 18 7 13 28 34 23 24 25 6 12 \ 20 27 33 31 32 11 19 26 36\\ 37 38 39 40 41 42 43 44 45 46 47 48 ), ( 1 2 43 4 45 6 7 48 9 10 11 12 13 3 17 23 32 18 19 20 5 16 31 24 25 26 27 \ 28 8 15 22 30 33 34 14 36\\ 21 38 39 29 41 42 35 44 37 46 47 40 ), ( 24 18 9 4 5 6 7 8 38 10 11 12 13 14 15 16 1 39 19 20 21 22 2 40 25 26 27 \ 28 29 30 31 3 33 34 35 36\\ 37 32 23 17 43 45 48 42 47 41 44 46 ), ( 12 2 3 20 5 27 7 8 11 19 26 33 13 14 15 16 17 10 25 36 21 22 23 9 18 24 \ 38 28 29 30 31 32 41 34 35\\ 44 37 46 39 40 1 42 43 4 45 6 47 48 ), ( 1 2 3 4 5 15 22 30 9 10 8 14 21 29 35 16 17 18 7 13 28 34 23 24 25 6 12 \ 20 27 33 31 32 11 19 26 36\\ 37 38 39 40 41 42 43 44 45 46 47 48 ), ( 1 2 43 4 45 6 7 48 9 10 11 12 13 3 17 23 32 18 19 20 5 16 31 24 25 26 27 \ 28 8 15 22 30 33 34 14 36\\ 21 38 39 29 41 42 35 44 37 46 47 40 ), ( 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 27 28 29 30 \ 31 32 43 42 41 35 37 40 34\\ 39 33 36 38 26 25 24 44 45 46 47 48 ), ( 1 2 43 4 45 6 7 48 9 10 11 12 13 3 17 23 32 18 19 20 5 16 31 24 25 26 27 \ 28 8 15 22 30 33 34 14 36\\ 21 38 39 29 41 42 35 44 37 46 47 40 ), ( 24 18 9 4 5 6 7 8 38 10 11 12 13 14 15 16 1 39 19 20 21 22 2 40 25 26 27 \ 28 29 30 31 3 33 34 35 36\\ 37 32 23 17 43 45 48 42 47 41 44 46 ), ( 24 18 9 4 5 6 7 8 38 10 11 12 13 14 15 16 1 39 19 20 21 22 2 40 25 26 27 \ 28 29 30 31 3 33 34 35 36\\ 37 32 23 17 43 45 48 42 47 41 44 46 ), ( 24 18 9 4 5 6 7 8 38 10 11 12 13 14 15 16 1 39 19 20 21 22 2 40 25 26 27 \ 28 29 30 31 3 33 34 35 36\\ 37 32 23 17 43 45 48 42 47 41 44 46 ), ( 12 2 3 20 5 27 7 8 11 19 26 33 13 14 15 16 17 10 25 36 21 22 23 9 18 24 \ 38 28 29 30 31 32 41 34 35\\ 44 37 46 39 40 1 42 43 4 45 6 47 48 ), ( 24 18 9 4 5 6 7 8 38 10 11 12 13 14 15 16 1 39 19 20 21 22 2 40 25 26 27 \ 28 29 30 31 3 33 34 35 36\\ 37 32 23 17 43 45 48 42 47 41 44 46 ), ( 1 2 43 4 45 6 7 48 9 10 11 12 13 3 17 23 32 18 19 20 5 16 31 24 25 26 27 \ 28 8 15 22 30 33 34 14 36\\ 21 38 39 29 41 42 35 44 37 46 47 40 ), ( 24 18 9 4 5 6 7 8 38 10 11 12 13 14 15 16 1 39 19 20 21 22 2 40 25 26 27 \ 28 29 30 31 3 33 34 35 36\\ 37 32 23 17 43 45 48 42 47 41 44 46 ), ( 1 2 43 4 45 6 7 48 9 10 11 12 13 3 17 23 32 18 19 20 5 16 31 24 25 26 27 \ 28 8 15 22 30 33 34 14 36\\ 21 38 39 29 41 42 35 44 37 46 47 40 ), ( 24 18 9 4 5 6 7 8 38 10 11 12 13 14 15 16 1 39 19 20 21 22 2 40 25 26 27 \ 28 29 30 31 3 33 34 35 36\\ 37 32 23 17 43 45 48 42 47 41 44 46 ), ( 24 18 9 4 5 6 7 8 38 10 11 12 13 14 15 16 1 39 19 20 21 22 2 40 25 26 27 \ 28 29 30 31 3 33 34 35 36\\ 37 32 23 17 43 45 48 42 47 41 44 46 ), ( 1 2 43 4 45 6 7 48 9 10 11 12 13 3 17 23 32 18 19 20 5 16 31 24 25 26 27 \ 28 8 15 22 30 33 34 14 36\\ 21 38 39 29 41 42 35 44 37 46 47 40 )}, {0, 0, 1, 7, 2, 8, 5, 3, 3, 18, 9, 1, 20, 12, 6, 22, 40, 2, 10, 4, 37, 7, \ 39, 1, 19, 11, 24, 21, 14, 8, 23, 38, 12, 22, 43, 39, 45, 9, 18, 24, 33, 44, 3, 47, 5, 48, 45, 8}}\ \>"], "Output"] }, Open ]], Cell["This is the set of orbits:", "Text"], Cell[CellGroupData[{ Cell["orbits[[1]]", "Input"], Cell[OutputFormData["\<\ HoldList[HoldList[1, 3, 12, 24, 8, 9, 43, 14, 33, 40, 27, 6, 30, 48, 11, 38, \ 35, 29, 41, 17, 15, 46, 26, 32], HoldList[2, 5, 18, 7, 45, 10, 39, 4, 22, 47, 37, 19, 23, 36, 20, 34, \ 16, 44, 21, 25, 31, 13, 42, 28]]\ \>", "\<\ {{1, 3, 12, 24, 8, 9, 43, 14, 33, 40, 27, 6, 30, 48, 11, 38, 35, 29, 41, 17, \ 15, 46, 26, 32}, {2, 5, 18, 7, 45, 10, 39, 4, 22, 47, 37, 19, 23, 36, 20, 34, 16, 44, 21, \ 25, 31, 13, 42, 28}}\ \>"], "Output"] }, Open ]], Cell["The group of Rubik's cube has the following two 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