### 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 [64,190]. 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 [64,190]. # 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(64,190));' can be # used to create a list e of the elements of the group [64,190]. # In our list of (64,28,12) DSs in group [64,190], we have [1, # 2, 3, 4, 5, 6, 8, 10, 18, 20, 21, 23, 26, 28, 29, # 31, 32, 34, 35, 38, 42, 45, 47, 49, 59, 61, 62, 64]. # This list denotes the elements [ e[1], e[2], e[3], # e[4], e[5], e[6], e[8], e[10], e[18], e[20], e[21], e[23], e[26], e[28], e[29], e[31], # e[32], e[34], e[35], e[38], e[42], e[45], e[47], e[49], e[59], e[61], e[62], e[64]] # where e is the list of the elements of the group [64,190] as they # appear when one uses the GAP command 'e:=Elements(SmallGroup(96,10))(64,190));' ########################### ########################### ########################### ### List of all (64,28,12) difference sets and the nonisomorphic symmetric (64,28,12) designs ### that rise from these DSs in group [64, 190] ### The number of inequivalent (64,28,12) DSs in group [64,190] is 88 ### The list of all inequivalent (64,28,12) DSs in group [64,190] is DS190:=[ [ 1, 2, 3, 4, 5, 6, 8, 10, 18, 20, 21, 23, 26, 28, 29, 31, 32, 34, 35, 38, 42, 45, 47, 49, 59, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 15, 18, 20, 21, 23, 26, 28, 29, 31, 32, 35, 42, 45, 46, 47, 49, 55, 58, 59, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 13, 16, 18, 20, 21, 23, 28, 29, 31, 32, 33, 35, 37, 42, 48, 49, 53, 56, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 13, 16, 18, 20, 21, 23, 25, 28, 29, 31, 32, 33, 35, 36, 37, 42, 44, 49, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 15, 18, 21, 23, 25, 32, 35, 39, 44, 46, 49, 50, 52, 55, 57, 58, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 15, 18, 21, 23, 25, 32, 35, 36, 39, 44, 49, 50, 52, 53, 55, 56, 57, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 15, 18, 21, 23, 24, 25, 32, 35, 39, 43, 44, 47, 49, 50, 52, 55, 57, 59 ], [ 1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 14, 15, 18, 21, 23, 25, 32, 33, 35, 37, 39, 44, 49, 50, 52, 54, 55, 57 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 14, 15, 16, 17, 20, 23, 26, 27, 28, 31, 32, 37, 38, 41, 45, 52, 53, 57, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 16, 17, 20, 26, 28, 30, 31, 32, 33, 35, 41, 44, 50, 53, 54, 57, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 16, 17, 20, 24, 26, 28, 30, 31, 32, 35, 41, 44, 46, 47, 50, 57, 60, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 14, 16, 17, 20, 26, 28, 30, 31, 32, 35, 36, 37, 41, 44, 50, 56, 57, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 13, 14, 16, 17, 20, 26, 27, 28, 31, 32, 35, 36, 37, 41, 44, 52, 56, 57, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 15, 18, 21, 23, 24, 27, 29, 31, 35, 39, 46, 47, 48, 49, 52, 55, 57, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 15, 18, 20, 21, 23, 24, 27, 29, 31, 35, 42, 47, 48, 49, 52, 55, 58, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 15, 18, 21, 23, 27, 29, 31, 35, 36, 37, 39, 48, 49, 52, 55, 56, 57, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 15, 18, 20, 21, 23, 27, 29, 31, 35, 37, 42, 48, 49, 52, 53, 55, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 16, 18, 20, 24, 25, 26, 27, 28, 31, 34, 40, 43, 47, 48, 49, 52, 55, 57, 59 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 16, 18, 20, 23, 26, 27, 28, 31, 34, 40, 43, 45, 49, 52, 55, 57, 58, 59, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 16, 18, 20, 23, 26, 27, 28, 31, 33, 34, 40, 45, 49, 52, 53, 54, 55, 57, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 16, 18, 20, 23, 24, 26, 27, 28, 31, 34, 40, 45, 46, 47, 49, 52, 55, 57, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 15, 16, 18, 20, 24, 26, 27, 28, 31, 38, 40, 43, 44, 47, 49, 52, 57, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 15, 16, 18, 20, 23, 24, 26, 27, 28, 31, 38, 40, 45, 47, 49, 52, 57, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 18, 20, 25, 26, 27, 28, 31, 33, 34, 37, 40, 48, 49, 52, 54, 55, 57 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 18, 20, 23, 26, 27, 28, 31, 34, 36, 37, 40, 45, 49, 52, 55, 56, 57 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 14, 15, 16, 18, 20, 26, 27, 28, 31, 33, 37, 38, 40, 44, 49, 52, 54, 57, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 14, 15, 16, 18, 20, 23, 26, 27, 28, 31, 37, 38, 40, 45, 49, 52, 53, 57, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 16, 18, 20, 26, 27, 28, 31, 35, 40, 43, 44, 49, 52, 57, 58, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 16, 18, 20, 26, 27, 28, 31, 33, 35, 40, 44, 49, 52, 53, 54, 57, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 16, 18, 20, 24, 26, 27, 28, 31, 35, 40, 44, 46, 47, 49, 52, 57, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 16, 18, 20, 24, 25, 26, 27, 28, 31, 35, 40, 47, 48, 49, 52, 57, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 14, 16, 18, 20, 26, 27, 28, 31, 35, 36, 37, 40, 44, 49, 52, 56, 57, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 11, 13, 14, 16, 18, 20, 25, 26, 27, 28, 31, 35, 37, 40, 48, 49, 52, 53, 57, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 16, 25, 26, 27, 28, 32, 34, 40, 41, 42, 46, 48, 49, 52, 55, 57, 58, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 16, 24, 25, 26, 27, 28, 32, 34, 40, 41, 42, 43, 47, 48, 49, 52, 55, 57, 59 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 16, 23, 24, 26, 27, 28, 32, 34, 40, 41, 42, 45, 46, 47, 49, 52, 55, 57, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 15, 18, 21, 30, 31, 32, 35, 39, 44, 45, 46, 48, 51, 52, 55, 57, 58, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 15, 18, 21, 30, 31, 32, 35, 36, 39, 44, 45, 48, 51, 52, 53, 55, 56, 57, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 15, 18, 21, 24, 30, 31, 32, 35, 39, 43, 44, 45, 47, 48, 51, 52, 55, 57, 59 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 15, 18, 20, 21, 31, 32, 35, 42, 44, 45, 46, 48, 49, 51, 55, 58, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 15, 18, 20, 21, 31, 32, 35, 36, 42, 44, 45, 48, 49, 51, 53, 55, 56, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 15, 18, 20, 21, 24, 31, 32, 35, 42, 43, 44, 45, 47, 48, 49, 51, 55, 59, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 15, 16, 26, 27, 28, 32, 38, 40, 41, 42, 44, 46, 49, 52, 57, 58, 60, 61, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 15, 16, 24, 26, 27, 28, 32, 38, 40, 41, 42, 43, 44, 47, 49, 52, 57, 59, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 15, 16, 23, 26, 27, 28, 32, 38, 40, 41, 42, 43, 45, 46, 49, 52, 57, 59, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 15, 16, 23, 24, 26, 27, 28, 32, 38, 40, 41, 42, 45, 47, 49, 52, 57, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 15, 16, 18, 26, 30, 32, 38, 39, 40, 42, 44, 46, 50, 51, 58, 60, 61, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 15, 16, 18, 26, 30, 32, 36, 38, 39, 40, 42, 44, 50, 51, 53, 56, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 14, 16, 25, 26, 28, 30, 32, 33, 34, 37, 40, 41, 42, 48, 49, 50, 54, 55, 57 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 14, 16, 18, 25, 26, 30, 32, 33, 34, 37, 39, 40, 42, 48, 50, 51, 54, 55, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 14, 15, 18, 21, 30, 31, 32, 33, 35, 37, 39, 44, 45, 48, 51, 52, 54, 55, 57 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 14, 15, 18, 20, 21, 31, 32, 33, 35, 37, 42, 44, 45, 48, 49, 51, 54, 55, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 14, 15, 16, 18, 26, 30, 32, 33, 37, 38, 39, 40, 42, 44, 50, 51, 54, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 26, 28, 30, 32, 35, 40, 41, 42, 43, 44, 49, 50, 57, 58, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 26, 28, 30, 32, 33, 35, 40, 41, 42, 44, 49, 50, 53, 54, 57, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 26, 27, 28, 32, 35, 40, 41, 42, 43, 44, 49, 52, 57, 58, 59, 60, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 26, 27, 28, 32, 33, 35, 40, 41, 42, 44, 49, 52, 53, 54, 57, 60, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 25, 26, 28, 30, 32, 35, 40, 41, 42, 43, 46, 48, 49, 50, 57, 59, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 25, 26, 28, 30, 32, 33, 35, 36, 40, 41, 42, 48, 49, 50, 54, 56, 57 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 25, 26, 27, 28, 32, 35, 40, 41, 42, 43, 46, 48, 49, 52, 57, 59, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 24, 26, 28, 30, 32, 35, 40, 41, 42, 44, 46, 47, 49, 50, 57, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 24, 26, 27, 28, 32, 35, 40, 41, 42, 44, 46, 47, 49, 52, 57, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 24, 25, 26, 28, 30, 32, 35, 40, 41, 42, 47, 48, 49, 50, 57, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 24, 25, 26, 27, 28, 32, 35, 40, 41, 42, 47, 48, 49, 52, 57, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 18, 26, 30, 32, 35, 39, 40, 42, 43, 44, 50, 51, 58, 59, 60, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 18, 26, 30, 32, 33, 35, 39, 40, 42, 44, 50, 51, 53, 54, 60, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 18, 25, 26, 30, 32, 35, 39, 40, 42, 43, 46, 48, 50, 51, 59, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 18, 25, 26, 30, 32, 33, 35, 36, 39, 40, 42, 48, 50, 51, 54, 56, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 18, 24, 26, 30, 32, 35, 39, 40, 42, 44, 46, 47, 50, 51, 60, 61, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 16, 18, 24, 25, 26, 30, 32, 35, 39, 40, 42, 47, 48, 50, 51, 58, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 14, 16, 26, 28, 30, 32, 35, 36, 37, 40, 41, 42, 44, 49, 50, 56, 57, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 14, 16, 26, 27, 28, 32, 35, 36, 37, 40, 41, 42, 44, 49, 52, 56, 57, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 14, 16, 25, 26, 28, 30, 32, 35, 37, 40, 41, 42, 48, 49, 50, 53, 57, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 14, 16, 25, 26, 27, 28, 32, 35, 37, 40, 41, 42, 48, 49, 52, 53, 57, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 14, 16, 18, 26, 30, 32, 35, 36, 37, 39, 40, 42, 44, 50, 51, 56, 60, 62 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 13, 14, 16, 18, 25, 26, 30, 32, 35, 37, 39, 40, 42, 48, 50, 51, 53, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 21, 25, 27, 29, 30, 33, 34, 37, 42, 45, 49, 53, 55, 56, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 21, 24, 25, 26, 27, 29, 35, 39, 46, 48, 52, 55, 57, 59, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 21, 24, 25, 26, 27, 29, 30, 35, 39, 48, 49, 55, 57, 58, 59, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 21, 23, 24, 25, 29, 35, 39, 44, 46, 47, 49, 50, 52, 55, 57, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 21, 24, 26, 27, 29, 35, 42, 44, 46, 52, 55, 59, 60, 62, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 21, 24, 26, 27, 29, 30, 35, 42, 44, 49, 55, 58, 59, 60, 61 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 21, 23, 24, 25, 29, 35, 42, 44, 47, 49, 50, 52, 55, 58, 64 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 21, 25, 27, 29, 30, 33, 37, 38, 39, 45, 49, 53, 56, 57, 60 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 16, 18, 20, 21, 27, 29, 34, 36, 42, 44, 45, 48, 52, 54, 55, 62, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 18, 21, 23, 25, 29, 35, 36, 37, 39, 44, 49, 50, 52, 55, 56, 57 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 18, 20, 21, 23, 25, 29, 35, 37, 42, 44, 49, 50, 52, 53, 55, 63 ], [ 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 21, 27, 29, 36, 38, 39, 44, 45, 48, 52, 54, 57, 62, 63 ] ]; ### The number of nonisomorphic symmetric (64,28,12) designs that rise from these DSs is 54 ### These nonisomorphic symmetric (64,28,12) designs are the ones that rise from the following DSs: ### DS190[1], DS190[2], DS190[3], DS190[4], DS190[5], DS190[6], DS190[7], DS190[8], DS190[14], DS190[15], DS190[16], DS190[17], DS190[21], DS190[26], DS190[27], DS190[33], DS190[36], DS190[37], DS190[38], DS190[39], DS190[40], DS190[41], DS190[42], DS190[44], DS190[45], DS190[47], DS190[48], DS190[49], DS190[51], DS190[52], DS190[54], DS190[55], DS190[56], DS190[61], DS190[63], DS190[65], DS190[67], DS190[68], DS190[71], DS190[72], DS190[73], DS190[75], DS190[77], DS190[78], DS190[79], DS190[80], DS190[81], DS190[82], DS190[83], DS190[84], DS190[85], DS190[86], DS190[87], DS190[88]. # Finally, We list all inequivalent reversible (64,28,12) difference # sets in group [64,190]. ### The number of inequivalent reversible (64,28,12) DSs in group [64,190] is 128 ### The list of all inequivalent reversible (64,28,12) DSs in group [64,190] is RDS190:=[ [ 2, 3, 8, 9, 10, 12, 13, 15, 17, 24, 25, 29, 31, 33, 36, 37, 39, 40, 43, 44, 45, 49, 53, 55, 57, 58, 61, 62 ], [ 2, 3, 8, 9, 10, 12, 13, 15, 17, 23, 24, 25, 29, 31, 33, 34, 36, 37, 39, 40, 44, 49, 57, 58, 59, 61, 62, 63 ], [ 2, 3, 8, 9, 10, 12, 13, 14, 17, 24, 25, 29, 31, 38, 39, 40, 43, 45, 49, 53, 54, 55, 56, 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