### This work done by Omar A. AbuGhneim from Jordan University (Email: o_abughneim@yahoo.com). # There are 267 groups of order 64. Among them, there are 259 groups that # admit (64,28,12) difference sets. The other 8 groups do not have any # (64,28,12) difference sets. We wrote programs using the software GAP # to construct all (64,28,12) difference sets in the 259 groups of order # 64 that admit (64,28,12) difference sets. # We refer to the groups as they appear in the SmallGroups library of # the software package GAP. For instance, when we work in groups of # order 64 and write [64,14] we mean group number 14 of order 64 in the # GAP library. # The 8 groups that do not admit (64,28,12) difference sets are # [64, 1], [64, 38], [64, 47], [64, 50], [64, 52], [64, 53], [64, 54], # and [64, 186]. Indeed, these groups are the ones that have either # the cyclic group of order 32 as a factor group or have the dihedral # group of order 32 as a factor group. # Also, we find all (64,28,12) difference sets that are reversible # i.e. all (64,28,12) difference sets D with D=D^(-1). There are 184 # groups of order 64 that have reversible (64,28,12) difference sets. # These groups are [64,i] where i in {2, 3, 4, 5, 8, 9, 10, 11, 12, 13, # 15, 16, 18, 20, 22, 23, 24, 25, 28, 30, 31, 32, 33, 34, 35, 36, 37, 40, # 41, 42, 43, 45, 46, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, # 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 84, 85, 87, # 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 104, 105, 106, # 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 126, # 127, 128, 129, 130, 131, 133, 134, 135, 138, 140, 141, 142, 144, 145, 146, # 147, 149, 150, 151, 152, 153, 155, 156, 157, 159, 160, 161, 162, 163, 164, # 166, 167, 169, 170, 172, 173, 174, 177, 178, 179, 182, 184, 187, 190, 192, # 193, 194, 195, 196, 197, 199, 200, 202, 203, 204, 205, 206, 207, 208, 209, # 210, 211, 213, 214, 215, 216, 219, 221, 222, 226, 227, 231, 232, 234, 236, # 240, 241, 242, 245, 247, 249, 250, 251, 254, 260, 261, 263, 267 }. # We used the software GAP to find all (64,28,12) difference sets and # reversible (64,28,12) difference sets. # In this file we list all inequivalent (64,28,12) difference sets # in group [64,167]. Also, we determine the nonisomorphic # symmetric (64,28,12) designs that rise from these DSs. # Finally, We list all inequivalent reversible (64,28,12) difference # sets in group [64,167]. # To determine the nonisomorphic symmetric (64,28,12) designs, we used # the DESIGN package for GAP, Version 1.6, 2011, by L.H. Soicher, # http://designtheory.org/software/gap design/. # Note: The GAP command 'e:=Elements(SmallGroup(64,167));' can be # used to create a list e of the elements of the group [64,167]. # In our list of (64,28,12) DSs in group [64,167], we have [1, # 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 17, 21, 23, 26, # 34, 35, 36, 38, 39, 41, 43, 44, 46, 52, 55, 61, 64]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[7], e[8], e[10], e[11], e[13], e[14], e[17], e[21], e[23], e[26], e[34], # e[35], e[36], e[38], e[39], e[41], e[43], e[44], e[46], e[52], e[55], e[61], e[64]] # where e is the list of the elements of the group [64,167] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,167));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 167] ### The number of inequivalent (64,28,12) DSs in group [64,167] is 214 ### The list of all inequivalent (64,28,12) DSs in group [64,167] is DS167:=[ [ 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 17, 21, 23, 26, 34, 35, 36, 38, 39, 41, 43, 44, 46, 52, 55, 61, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 12, 13, 14, 17, 21, 24, 25, 30, 34, 35, 36, 38, 39, 41, 50, 51, 52, 55, 61, 62 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 11, 14, 15, 21, 23, 25, 26, 29, 30, 31, 40, 41, 43, 48, 49, 56, 57, 58, 60, 62 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 15, 18, 21, 24, 25, 32, 35, 36, 37, 39, 40, 50, 51, 53, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 14, 17, 21, 24, 25, 32, 35, 36, 38, 39, 41, 50, 51, 53, 61, 62, 63 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 10, 14, 15, 21, 23, 25, 26, 29, 30, 32, 40, 41, 44, 48, 49, 56, 57, 58, 59, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 14, 17, 26, 32, 33, 34, 36, 38, 39, 41, 42, 44, 46, 58, 60, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 14, 17, 23, 32, 33, 35, 36, 38, 39, 41, 42, 45, 47, 48, 54, 58, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 14, 17, 23, 26, 32, 33, 34, 36, 38, 39, 41, 42, 43, 45, 46, 59, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 11, 12, 13, 14, 17, 23, 24, 26, 32, 33, 34, 36, 38, 39, 41, 42, 45, 47, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 13, 15, 16, 19, 23, 24, 26, 30, 32, 33, 34, 38, 39, 42, 44, 47, 57, 58, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 13, 14, 16, 19, 23, 25, 26, 30, 32, 33, 34, 37, 39, 42, 44, 48, 57, 58, 59, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 13, 14, 16, 18, 19, 23, 26, 30, 32, 33, 34, 37, 41, 42, 44, 46, 58, 59, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 14, 17, 26, 31, 33, 34, 36, 38, 39, 41, 42, 43, 46, 58, 59, 61, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 14, 17, 23, 31, 34, 35, 36, 38, 39, 41, 42, 45, 47, 48, 55, 58, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 14, 17, 23, 26, 31, 33, 34, 36, 38, 39, 41, 42, 44, 45, 46, 60, 61, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 14, 17, 23, 25, 26, 31, 33, 34, 36, 38, 39, 41, 42, 45, 48, 58, 63, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 12, 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27, 38, 39, 40, 42, 45, 46, 51, 54, 55, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 16, 17, 20, 23, 24, 25, 27, 33, 34, 38, 39, 40, 42, 45, 46, 51, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 17, 25, 29, 30, 35, 36, 38, 39, 41, 48, 50, 51, 52, 54, 55, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 17, 25, 29, 30, 33, 34, 35, 36, 38, 39, 41, 48, 50, 51, 52, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 17, 24, 29, 30, 35, 36, 38, 39, 41, 47, 50, 51, 52, 54, 55, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 17, 24, 29, 30, 33, 34, 35, 36, 38, 39, 41, 47, 50, 51, 52, 61 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 17, 24, 25, 26, 29, 30, 34, 36, 38, 39, 41, 50, 51, 52, 55, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 17, 24, 25, 26, 29, 30, 33, 36, 38, 39, 41, 50, 51, 52, 54, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 17, 24, 25, 26, 28, 30, 35, 36, 38, 39, 41, 51, 52, 54, 55, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 17, 24, 25, 26, 28, 30, 33, 34, 35, 36, 38, 39, 41, 51, 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9, 10, 15, 16, 17, 19, 20, 23, 30, 32, 35, 38, 40, 42, 44, 53, 58, 59, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 16, 18, 19, 20, 23, 30, 32, 35, 37, 41, 42, 44, 53, 58, 59, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 26, 29, 30, 32, 33, 35, 37, 38, 43, 45, 53, 54, 56, 57, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 23, 26, 29, 30, 32, 33, 35, 37, 38, 44, 53, 54, 56, 57, 58, 59 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 21, 22, 23, 26, 29, 30, 32, 33, 40, 41, 44, 54, 56, 57, 58, 59 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 20, 23, 25, 28, 31, 35, 42, 44, 45, 46, 47, 49, 50, 53, 56, 57, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 20, 23, 24, 27, 31, 35, 42, 44, 45, 46, 48, 49, 51, 53, 56, 57, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 18, 20, 32, 35, 37, 39, 41, 42, 43, 53, 58, 60, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 18, 19, 20, 32, 35, 36, 37, 40, 42, 44, 53, 58, 59, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 17, 20, 23, 31, 35, 36, 38, 39, 41, 42, 44, 45, 53, 60, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 32, 35, 38, 39, 41, 42, 44, 53, 58, 59, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 23, 32, 35, 38, 39, 41, 42, 44, 45, 53, 59, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 17, 20, 32, 35, 38, 39, 40, 42, 44, 53, 58, 59, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 17, 20, 23, 32, 35, 38, 39, 40, 42, 44, 45, 53, 59, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 21, 22, 23, 26, 29, 30, 34, 37, 39, 40, 41, 43, 44, 55, 56, 58 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 21, 22, 23, 26, 29, 30, 33, 37, 39, 40, 41, 43, 44, 54, 56, 58 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 21, 22, 26, 29, 30, 34, 38, 39, 40, 41, 43, 44, 45, 55, 56, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 21, 22, 26, 29, 30, 33, 38, 39, 40, 41, 43, 44, 45, 54, 56, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 17, 21, 22, 23, 26, 29, 34, 36, 38, 39, 41, 43, 44, 52, 55, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 17, 21, 22, 23, 26, 29, 33, 36, 38, 39, 41, 43, 44, 52, 54, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 18, 21, 22, 26, 29, 30, 34, 36, 40, 45, 55, 57, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 18, 21, 22, 26, 29, 30, 34, 36, 40, 43, 44, 45, 55, 57, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 18, 21, 22, 26, 29, 30, 33, 36, 40, 45, 54, 57, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 18, 21, 22, 26, 29, 30, 33, 36, 40, 43, 44, 45, 54, 57, 64 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 17, 21, 22, 23, 26, 29, 30, 34, 36, 41, 55, 57, 58, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 17, 21, 22, 23, 26, 29, 30, 34, 36, 41, 43, 44, 55, 57, 58 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 17, 21, 22, 23, 26, 29, 30, 33, 36, 41, 54, 57, 58, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14, 15, 17, 21, 22, 23, 26, 29, 30, 33, 36, 41, 43, 44, 54, 57, 58 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 26, 29, 30, 34, 36, 38, 39, 41, 45, 52, 55, 58, 63 ], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 17, 26, 29, 30, 33, 36, 38, 39, 41, 45, 52, 54, 58, 63 ] ]; ### The number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 154 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS167[1], DS167[6], DS167[7], DS167[8], DS167[10], DS167[12], DS167[15], DS167[18], DS167[19], DS167[20], DS167[21], DS167[25], DS167[26], DS167[29], DS167[30], DS167[32], DS167[33], DS167[34], DS167[35], DS167[38], DS167[40], DS167[41], DS167[43], DS167[44], DS167[48], DS167[49], DS167[50], DS167[53], DS167[56], DS167[61], DS167[62], DS167[63], DS167[64], DS167[65], DS167[70], DS167[71], DS167[72], DS167[73], DS167[75], DS167[76], DS167[77], DS167[78], DS167[79], DS167[80], DS167[82], DS167[83], DS167[84], DS167[85], DS167[86], DS167[87], DS167[88], DS167[89], DS167[90], DS167[92], DS167[93], DS167[95], DS167[96], DS167[97], DS167[98], DS167[99], DS167[100], DS167[102], DS167[104], DS167[105], DS167[106], DS167[107], DS167[110], DS167[111], DS167[115], DS167[116], DS167[117], DS167[118], DS167[119], DS167[120], DS167[121], DS167[122], DS167[123], DS167[124], DS167[126], DS167[127], DS167[128], DS167[129], DS167[130], DS167[131], DS167[132], DS167[133], DS167[134], DS167[135], DS167[136], DS167[137], DS167[138], DS167[139], DS167[140], DS167[141], DS167[144], DS167[145], DS167[146], DS167[147], DS167[148], DS167[149], DS167[150], DS167[151], DS167[152], DS167[153], DS167[154], DS167[155], DS167[156], DS167[158], DS167[160], DS167[161], DS167[162], DS167[163], DS167[164], DS167[165], DS167[166], DS167[167], DS167[168], DS167[169], DS167[170], DS167[171], DS167[172], DS167[173], DS167[174], DS167[175], DS167[176], DS167[177], DS167[178], DS167[180], DS167[181], DS167[187], DS167[188], DS167[189], DS167[191], DS167[192], DS167[193], DS167[194], DS167[195], DS167[196], DS167[197], DS167[198], DS167[199], DS167[200], DS167[201], DS167[202], DS167[203], DS167[205], DS167[206], DS167[207], DS167[208], DS167[209], DS167[210], DS167[211], DS167[212], DS167[213]. # Finally, We list all inequivalent reversible (64,28,12) difference # sets in group [64,167]. ### The number of inequivalent reversible (64,28,12) DSs in group [64,167] is 72 ### The list of all inequivalent reversible (64,28,12) DSs in group [64,167] is RDS167:=[ [ 2, 3, 6, 8, 9, 10, 11, 20, 22, 23, 24, 27, 33, 34, 37, 38, 42, 44, 45, 46, 48, 49, 51, 52, 54, 55, 56, 60 ], [ 2, 3, 6, 8, 9, 10, 11, 15, 16, 20, 22, 23, 24, 27, 33, 34, 38, 42, 43, 44, 46, 48, 49, 51, 52, 54, 55, 64 ], [ 2, 3, 6, 8, 9, 10, 11, 13, 20, 22, 23, 24, 27, 35, 37, 38, 42, 44, 45, 46, 48, 49, 51, 52, 53, 56, 60, 63 ], [ 2, 3, 6, 8, 9, 10, 11, 13, 15, 16, 20, 22, 23, 24, 27, 35, 38, 42, 43, 44, 46, 48, 49, 51, 52, 53, 63, 64 ], [ 2, 3, 5, 8, 9, 10, 11, 20, 21, 23, 24, 27, 33, 34, 37, 38, 42, 43, 45, 46, 48, 49, 51, 52, 54, 55, 56, 59 ], [ 2, 3, 5, 8, 9, 10, 11, 13, 20, 21, 23, 24, 27, 35, 37, 38, 42, 43, 45, 46, 48, 49, 51, 52, 53, 56, 59, 63 ], [ 2, 3, 4, 8, 9, 10, 11, 15, 16, 23, 25, 28, 33, 34, 38, 40, 41, 43, 45, 46, 47, 49, 50, 52, 54, 55, 57, 59 ], [ 2, 3, 4, 8, 9, 10, 11, 15, 16, 23, 24, 27, 33, 34, 38, 40, 41, 43, 45, 46, 48, 49, 51, 52, 54, 55, 57, 59 ], [ 2, 3, 4, 8, 9, 10, 11, 15, 16, 17, 19, 24, 27, 33, 34, 38, 40, 45, 46, 48, 49, 51, 52, 54, 55, 58, 59, 60 ], [ 2, 3, 4, 8, 9, 10, 11, 15, 16, 17, 19, 23, 24, 27, 33, 34, 38, 40, 43, 44, 46, 48, 49, 51, 52, 54, 55, 64 ], [ 2, 3, 4, 8, 9, 10, 11, 13, 15, 16, 23, 25, 28, 35, 38, 40, 41, 43, 45, 46, 47, 49, 50, 52, 53, 57, 59, 63 ], [ 2, 3, 4, 8, 9, 10, 11, 13, 15, 16, 23, 24, 27, 35, 38, 40, 41, 43, 45, 46, 48, 49, 51, 52, 53, 57, 59, 63 ], [ 2, 3, 4, 8, 9, 10, 11, 13, 15, 16, 17, 19, 24, 27, 35, 38, 40, 45, 46, 48, 49, 51, 52, 53, 58, 59, 60, 63 ], [ 2, 3, 4, 8, 9, 10, 11, 13, 15, 16, 17, 19, 23, 24, 27, 35, 38, 40, 43, 44, 46, 48, 49, 51, 52, 53, 63, 64 ], [ 2, 3, 4, 7, 8, 10, 11, 13, 17, 19, 23, 28, 33, 34, 37, 38, 40, 42, 44, 45, 46, 50, 52, 53, 54, 55, 56, 60 ], [ 2, 3, 4, 7, 8, 10, 11, 13, 15, 16, 28, 33, 34, 38, 40, 41, 42, 44, 46, 50, 52, 53, 54, 55, 57, 58, 60, 64 ], [ 2, 3, 4, 7, 8, 10, 11, 13, 15, 16, 23, 28, 33, 34, 38, 40, 41, 42, 43, 45, 46, 50, 52, 53, 54, 55, 57, 59 ], [ 2, 3, 4, 7, 8, 10, 11, 13, 15, 16, 17, 19, 23, 28, 33, 34, 38, 40, 42, 43, 44, 46, 50, 52, 53, 54, 55, 64 ], [ 2, 3, 4, 7, 8, 9, 13, 15, 16, 17, 19, 20, 24, 27, 33, 34, 35, 38, 40, 43, 44, 46, 48, 49, 51, 52, 53, 63 ], [ 2, 3, 4, 7, 8, 9, 10, 12, 13, 15, 16, 28, 29, 31, 38, 40, 41, 42, 45, 46, 49, 50, 53, 57, 58, 59, 60, 62 ], [ 2, 3, 4, 7, 8, 9, 10, 12, 13, 15, 16, 23, 28, 29, 31, 38, 40, 41, 42, 43, 44, 46, 49, 50, 53, 57, 62, 64 ], [ 2, 3, 4, 7, 8, 9, 10, 12, 13, 15, 16, 23, 24, 26, 31, 38, 40, 41, 42, 43, 44, 46, 48, 49, 53, 57, 61, 64 ], [ 2, 3, 4, 7, 8, 9, 10, 11, 13, 15, 16, 28, 29, 38, 40, 41, 42, 44, 46, 49, 50, 52, 53, 57, 58, 60, 62, 64 ], [ 2, 3, 4, 7, 8, 9, 10, 11, 13, 15, 16, 23, 28, 29, 38, 40, 41, 42, 43, 45, 46, 49, 50, 52, 53, 57, 59, 62 ], [ 2, 3, 4, 7, 8, 9, 10, 11, 13, 15, 16, 23, 24, 26, 38, 40, 41, 42, 43, 45, 46, 48, 49, 52, 53, 57, 59, 61 ], [ 2, 3, 4, 7, 8, 9, 10, 11, 13, 15, 16, 17, 19, 23, 24, 26, 38, 40, 42, 43, 44, 46, 48, 49, 52, 53, 61, 64 ], [ 2, 3, 4, 7, 8, 9, 10, 11, 12, 15, 16, 17, 19, 20, 23, 24, 27, 30, 38, 40, 46, 48, 49, 51, 52, 54, 55, 64 ], [ 2, 3, 4, 5, 8, 9, 10, 11, 17, 19, 20, 21, 24, 27, 33, 34, 37, 38, 40, 42, 46, 48, 49, 51, 52, 54, 55, 56 ], [ 2, 3, 4, 5, 8, 9, 10, 11, 17, 19, 20, 21, 23, 25, 28, 37, 38, 40, 42, 45, 46, 47, 49, 50, 52, 56, 58, 64 ], [ 2, 3, 4, 5, 8, 9, 10, 11, 17, 19, 20, 21, 23, 24, 27, 37, 38, 40, 42, 45, 46, 48, 49, 51, 52, 56, 58, 64 ], [ 2, 3, 4, 5, 8, 9, 10, 11, 13, 17, 19, 20, 21, 24, 27, 35, 37, 38, 40, 42, 46, 48, 49, 51, 52, 53, 56, 63 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 11, 15, 16, 21, 22, 23, 25, 28, 38, 40, 41, 43, 45, 46, 47, 49, 50, 52, 57, 59 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 11, 15, 16, 21, 22, 23, 24, 27, 38, 40, 41, 43, 45, 46, 48, 49, 51, 52, 57, 59 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 11, 15, 16, 17, 19, 21, 22, 24, 27, 38, 40, 45, 46, 48, 49, 51, 52, 58, 59, 60 ], [ 2, 3, 4, 5, 6, 8, 9, 10, 11, 15, 16, 17, 19, 21, 22, 23, 24, 27, 38, 40, 43, 44, 46, 48, 49, 51, 52, 64 ], [ 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 17, 19, 21, 22, 23, 28, 37, 38, 40, 42, 44, 45, 46, 50, 52, 53, 56, 60 ], [ 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 15, 16, 21, 22, 28, 38, 40, 41, 42, 44, 46, 50, 52, 53, 57, 58, 60, 64 ], [ 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 15, 16, 21, 22, 23, 28, 38, 40, 41, 42, 43, 45, 46, 50, 52, 53, 57, 59 ], [ 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 15, 16, 17, 19, 21, 22, 23, 28, 38, 40, 42, 43, 44, 46, 50, 52, 53, 64 ], [ 2, 3, 4, 5, 6, 7, 8, 9, 15, 16, 17, 19, 20, 21, 22, 24, 27, 33, 34, 38, 40, 43, 44, 46, 48, 49, 51, 52 ], [ 1, 2, 3, 6, 7, 8, 9, 10, 11, 22, 23, 24, 27, 33, 34, 37, 38, 43, 45, 46, 48, 49, 51, 52, 54, 55, 56, 59 ], [ 1, 2, 3, 6, 7, 8, 9, 10, 11, 13, 22, 23, 24, 27, 35, 37, 38, 43, 45, 46, 48, 49, 51, 52, 53, 56, 59, 63 ], [ 1, 2, 3, 5, 7, 8, 9, 10, 11, 21, 23, 24, 27, 33, 34, 37, 38, 44, 45, 46, 48, 49, 51, 52, 54, 55, 56, 60 ], [ 1, 2, 3, 5, 7, 8, 9, 10, 11, 15, 16, 21, 23, 24, 27, 33, 34, 38, 43, 44, 46, 48, 49, 51, 52, 54, 55, 64 ], [ 1, 2, 3, 5, 7, 8, 9, 10, 11, 13, 21, 23, 24, 27, 35, 37, 38, 44, 45, 46, 48, 49, 51, 52, 53, 56, 60, 63 ], [ 1, 2, 3, 5, 7, 8, 9, 10, 11, 13, 15, 16, 21, 23, 24, 27, 35, 38, 43, 44, 46, 48, 49, 51, 52, 53, 63, 64 ], [ 1, 2, 3, 4, 8, 10, 11, 13, 17, 19, 20, 23, 27, 33, 34, 37, 38, 40, 44, 45, 46, 51, 52, 53, 54, 55, 56, 60 ], [ 1, 2, 3, 4, 8, 10, 11, 13, 15, 16, 20, 27, 33, 34, 38, 40, 41, 44, 46, 51, 52, 53, 54, 55, 57, 58, 60, 64 ], [ 1, 2, 3, 4, 8, 10, 11, 13, 15, 16, 20, 23, 27, 33, 34, 38, 40, 41, 43, 45, 46, 51, 52, 53, 54, 55, 57, 59 ], [ 1, 2, 3, 4, 8, 10, 11, 13, 15, 16, 17, 19, 20, 23, 27, 33, 34, 38, 40, 43, 44, 46, 51, 52, 53, 54, 55, 64 ], [ 1, 2, 3, 4, 8, 9, 13, 15, 16, 17, 19, 24, 27, 33, 34, 35, 38, 40, 42, 46, 48, 49, 51, 52, 53, 59, 60, 63 ], [ 1, 2, 3, 4, 8, 9, 10, 12, 13, 15, 16, 20, 27, 29, 31, 38, 40, 41, 45, 46, 49, 51, 53, 57, 58, 59, 60, 62 ], [ 1, 2, 3, 4, 8, 9, 10, 12, 13, 15, 16, 20, 23, 27, 29, 31, 38, 40, 41, 43, 44, 46, 49, 51, 53, 57, 62, 64 ], [ 1, 2, 3, 4, 8, 9, 10, 12, 13, 15, 16, 20, 23, 25, 26, 31, 38, 40, 41, 43, 44, 46, 47, 49, 53, 57, 61, 64 ], [ 1, 2, 3, 4, 8, 9, 10, 11, 13, 15, 16, 20, 27, 29, 38, 40, 41, 44, 46, 49, 51, 52, 53, 57, 58, 60, 62, 64 ], [ 1, 2, 3, 4, 8, 9, 10, 11, 13, 15, 16, 20, 23, 27, 29, 38, 40, 41, 43, 45, 46, 49, 51, 52, 53, 57, 59, 62 ], [ 1, 2, 3, 4, 8, 9, 10, 11, 13, 15, 16, 20, 23, 25, 26, 38, 40, 41, 43, 45, 46, 47, 49, 52, 53, 57, 59, 61 ], [ 1, 2, 3, 4, 8, 9, 10, 11, 13, 15, 16, 17, 19, 20, 23, 25, 26, 38, 40, 43, 44, 46, 47, 49, 52, 53, 61, 64 ], [ 1, 2, 3, 4, 8, 9, 10, 11, 12, 15, 16, 17, 19, 23, 24, 27, 30, 33, 34, 38, 40, 42, 46, 48, 49, 51, 52, 64 ], [ 1, 2, 3, 4, 7, 8, 9, 10, 11, 15, 16, 20, 23, 25, 28, 38, 40, 41, 42, 43, 45, 46, 47, 49, 50, 52, 57, 59 ], [ 1, 2, 3, 4, 7, 8, 9, 10, 11, 15, 16, 20, 23, 24, 27, 38, 40, 41, 42, 43, 45, 46, 48, 49, 51, 52, 57, 59 ], [ 1, 2, 3, 4, 7, 8, 9, 10, 11, 15, 16, 17, 19, 20, 24, 27, 38, 40, 42, 45, 46, 48, 49, 51, 52, 58, 59, 60 ], [ 1, 2, 3, 4, 7, 8, 9, 10, 11, 15, 16, 17, 19, 20, 23, 24, 27, 38, 40, 42, 43, 44, 46, 48, 49, 51, 52, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 21, 24, 27, 33, 34, 38, 40, 41, 46, 48, 49, 51, 52, 54, 55, 57 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 21, 23, 25, 28, 38, 40, 41, 45, 46, 47, 49, 50, 52, 57, 58, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 21, 23, 24, 27, 38, 40, 41, 45, 46, 48, 49, 51, 52, 57, 58, 64 ], [ 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 15, 16, 21, 24, 27, 35, 38, 40, 41, 46, 48, 49, 51, 52, 53, 57, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 13, 17, 19, 20, 21, 22, 23, 27, 37, 38, 40, 44, 45, 46, 51, 52, 53, 56, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 13, 15, 16, 20, 21, 22, 27, 38, 40, 41, 44, 46, 51, 52, 53, 57, 58, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 13, 15, 16, 20, 21, 22, 23, 27, 38, 40, 41, 43, 45, 46, 51, 52, 53, 57, 59 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 13, 15, 16, 17, 19, 20, 21, 22, 23, 27, 38, 40, 43, 44, 46, 51, 52, 53, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 15, 16, 17, 19, 21, 22, 24, 27, 33, 34, 38, 40, 42, 46, 48, 49, 51, 52, 59, 60 ] ]; ########################### ########################### ###########################