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## G = Dic5.SD16order 320 = 26·5

### 6th non-split extension by Dic5 of SD16 acting via SD16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — Dic5.SD16
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — Dic5⋊C8 — Dic5.SD16
 Lower central C5 — C2×C10 — C2×C20 — Dic5.SD16
 Upper central C1 — C22 — C2×C4 — C2×D4

Generators and relations for Dic5.SD16
G = < a,b,c,d | a10=c8=d2=1, b2=a5, bab-1=a-1, cac-1=a3, ad=da, cbc-1=dbd=a5b, dcd=bc3 >

Subgroups: 490 in 84 conjugacy classes, 26 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C10, C42, C2×C8, C2×D4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C4⋊C8, C41D4, C5⋊C8, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C4.D8, C4×Dic5, C2×C5⋊C8, C2×D20, C2×C5⋊D4, D4×C10, Dic5⋊C8, C20⋊D4, Dic5.SD16
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, D8, SD16, F5, C4.D4, D4⋊C4, C2×F5, C4.D8, C22⋊F5, D20⋊C4, C23.F5, Dic5.SD16

Character table of Dic5.SD16

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 5 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 10D 10E 10F 10G 20A 20B size 1 1 1 1 8 40 4 10 10 10 10 4 20 20 20 20 20 20 20 20 4 4 4 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ3 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ4 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 -i i i -i -i i i -i 1 1 1 1 1 1 1 1 1 linear of order 4 ρ6 1 1 1 1 -1 1 1 -1 -1 -1 -1 1 -i -i -i i i i i -i 1 1 1 -1 -1 -1 -1 1 1 linear of order 4 ρ7 1 1 1 1 -1 1 1 -1 -1 -1 -1 1 i i i -i -i -i -i i 1 1 1 -1 -1 -1 -1 1 1 linear of order 4 ρ8 1 1 1 1 1 -1 1 -1 -1 -1 -1 1 i -i -i i i -i -i i 1 1 1 1 1 1 1 1 1 linear of order 4 ρ9 2 2 2 2 0 0 -2 2 -2 -2 2 2 0 0 0 0 0 0 0 0 2 2 2 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ10 2 2 2 2 0 0 -2 -2 2 2 -2 2 0 0 0 0 0 0 0 0 2 2 2 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ11 2 -2 2 -2 0 0 0 0 2 -2 0 2 -√2 0 0 0 0 √2 -√2 √2 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D8 ρ12 2 -2 -2 2 0 0 0 2 0 0 -2 2 0 √2 -√2 -√2 √2 0 0 0 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D8 ρ13 2 -2 -2 2 0 0 0 2 0 0 -2 2 0 -√2 √2 √2 -√2 0 0 0 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D8 ρ14 2 -2 2 -2 0 0 0 0 2 -2 0 2 √2 0 0 0 0 -√2 √2 -√2 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D8 ρ15 2 -2 -2 2 0 0 0 -2 0 0 2 2 0 √-2 -√-2 √-2 -√-2 0 0 0 -2 2 -2 0 0 0 0 0 0 complex lifted from SD16 ρ16 2 -2 -2 2 0 0 0 -2 0 0 2 2 0 -√-2 √-2 -√-2 √-2 0 0 0 -2 2 -2 0 0 0 0 0 0 complex lifted from SD16 ρ17 2 -2 2 -2 0 0 0 0 -2 2 0 2 √-2 0 0 0 0 √-2 -√-2 -√-2 2 -2 -2 0 0 0 0 0 0 complex lifted from SD16 ρ18 2 -2 2 -2 0 0 0 0 -2 2 0 2 -√-2 0 0 0 0 -√-2 √-2 √-2 2 -2 -2 0 0 0 0 0 0 complex lifted from SD16 ρ19 4 4 4 4 4 0 4 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ20 4 4 -4 -4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 -4 -4 4 0 0 0 0 0 0 orthogonal lifted from C4.D4 ρ21 4 4 4 4 -4 0 4 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 1 1 -1 -1 orthogonal lifted from C2×F5 ρ22 4 4 4 4 0 0 -4 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 -√5 -√5 √5 √5 1 1 orthogonal lifted from C22⋊F5 ρ23 4 4 4 4 0 0 -4 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 -1 -1 √5 √5 -√5 -√5 1 1 orthogonal lifted from C22⋊F5 ρ24 4 4 -4 -4 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 -1 2ζ53+2ζ5+1 2ζ54+2ζ52+1 2ζ52+2ζ5+1 2ζ54+2ζ53+1 √5 -√5 complex lifted from C23.F5 ρ25 4 4 -4 -4 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 -1 2ζ52+2ζ5+1 2ζ54+2ζ53+1 2ζ54+2ζ52+1 2ζ53+2ζ5+1 -√5 √5 complex lifted from C23.F5 ρ26 4 4 -4 -4 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 -1 2ζ54+2ζ52+1 2ζ53+2ζ5+1 2ζ54+2ζ53+1 2ζ52+2ζ5+1 √5 -√5 complex lifted from C23.F5 ρ27 4 4 -4 -4 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 1 -1 2ζ54+2ζ53+1 2ζ52+2ζ5+1 2ζ53+2ζ5+1 2ζ54+2ζ52+1 -√5 √5 complex lifted from C23.F5 ρ28 8 -8 8 -8 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 orthogonal lifted from D20⋊C4, Schur index 2 ρ29 8 -8 -8 8 0 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 0 2 -2 2 0 0 0 0 0 0 orthogonal lifted from D20⋊C4, Schur index 2

Smallest permutation representation of Dic5.SD16
On 160 points
Generators in S160
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 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 46 47 48 49 50)(51 52 53 54 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 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160)
(1 89 6 84)(2 88 7 83)(3 87 8 82)(4 86 9 81)(5 85 10 90)(11 107 16 102)(12 106 17 101)(13 105 18 110)(14 104 19 109)(15 103 20 108)(21 79 26 74)(22 78 27 73)(23 77 28 72)(24 76 29 71)(25 75 30 80)(31 70 36 65)(32 69 37 64)(33 68 38 63)(34 67 39 62)(35 66 40 61)(41 60 46 55)(42 59 47 54)(43 58 48 53)(44 57 49 52)(45 56 50 51)(91 159 96 154)(92 158 97 153)(93 157 98 152)(94 156 99 151)(95 155 100 160)(111 146 116 141)(112 145 117 150)(113 144 118 149)(114 143 119 148)(115 142 120 147)(121 136 126 131)(122 135 127 140)(123 134 128 139)(124 133 129 138)(125 132 130 137)
(1 133 51 97 24 145 62 110)(2 140 60 100 25 142 61 103)(3 137 59 93 26 149 70 106)(4 134 58 96 27 146 69 109)(5 131 57 99 28 143 68 102)(6 138 56 92 29 150 67 105)(7 135 55 95 30 147 66 108)(8 132 54 98 21 144 65 101)(9 139 53 91 22 141 64 104)(10 136 52 94 23 148 63 107)(11 90 121 44 151 77 119 33)(12 87 130 47 152 74 118 36)(13 84 129 50 153 71 117 39)(14 81 128 43 154 78 116 32)(15 88 127 46 155 75 115 35)(16 85 126 49 156 72 114 38)(17 82 125 42 157 79 113 31)(18 89 124 45 158 76 112 34)(19 86 123 48 159 73 111 37)(20 83 122 41 160 80 120 40)
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 61)(11 100)(12 91)(13 92)(14 93)(15 94)(16 95)(17 96)(18 97)(19 98)(20 99)(21 58)(22 59)(23 60)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 81)(32 82)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 77)(42 78)(43 79)(44 80)(45 71)(46 72)(47 73)(48 74)(49 75)(50 76)(101 159)(102 160)(103 151)(104 152)(105 153)(106 154)(107 155)(108 156)(109 157)(110 158)(111 144)(112 145)(113 146)(114 147)(115 148)(116 149)(117 150)(118 141)(119 142)(120 143)(121 140)(122 131)(123 132)(124 133)(125 134)(126 135)(127 136)(128 137)(129 138)(130 139)

G:=sub<Sym(160)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,89,6,84)(2,88,7,83)(3,87,8,82)(4,86,9,81)(5,85,10,90)(11,107,16,102)(12,106,17,101)(13,105,18,110)(14,104,19,109)(15,103,20,108)(21,79,26,74)(22,78,27,73)(23,77,28,72)(24,76,29,71)(25,75,30,80)(31,70,36,65)(32,69,37,64)(33,68,38,63)(34,67,39,62)(35,66,40,61)(41,60,46,55)(42,59,47,54)(43,58,48,53)(44,57,49,52)(45,56,50,51)(91,159,96,154)(92,158,97,153)(93,157,98,152)(94,156,99,151)(95,155,100,160)(111,146,116,141)(112,145,117,150)(113,144,118,149)(114,143,119,148)(115,142,120,147)(121,136,126,131)(122,135,127,140)(123,134,128,139)(124,133,129,138)(125,132,130,137), (1,133,51,97,24,145,62,110)(2,140,60,100,25,142,61,103)(3,137,59,93,26,149,70,106)(4,134,58,96,27,146,69,109)(5,131,57,99,28,143,68,102)(6,138,56,92,29,150,67,105)(7,135,55,95,30,147,66,108)(8,132,54,98,21,144,65,101)(9,139,53,91,22,141,64,104)(10,136,52,94,23,148,63,107)(11,90,121,44,151,77,119,33)(12,87,130,47,152,74,118,36)(13,84,129,50,153,71,117,39)(14,81,128,43,154,78,116,32)(15,88,127,46,155,75,115,35)(16,85,126,49,156,72,114,38)(17,82,125,42,157,79,113,31)(18,89,124,45,158,76,112,34)(19,86,123,48,159,73,111,37)(20,83,122,41,160,80,120,40), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,61)(11,100)(12,91)(13,92)(14,93)(15,94)(16,95)(17,96)(18,97)(19,98)(20,99)(21,58)(22,59)(23,60)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,81)(32,82)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,77)(42,78)(43,79)(44,80)(45,71)(46,72)(47,73)(48,74)(49,75)(50,76)(101,159)(102,160)(103,151)(104,152)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,144)(112,145)(113,146)(114,147)(115,148)(116,149)(117,150)(118,141)(119,142)(120,143)(121,140)(122,131)(123,132)(124,133)(125,134)(126,135)(127,136)(128,137)(129,138)(130,139)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (1,89,6,84)(2,88,7,83)(3,87,8,82)(4,86,9,81)(5,85,10,90)(11,107,16,102)(12,106,17,101)(13,105,18,110)(14,104,19,109)(15,103,20,108)(21,79,26,74)(22,78,27,73)(23,77,28,72)(24,76,29,71)(25,75,30,80)(31,70,36,65)(32,69,37,64)(33,68,38,63)(34,67,39,62)(35,66,40,61)(41,60,46,55)(42,59,47,54)(43,58,48,53)(44,57,49,52)(45,56,50,51)(91,159,96,154)(92,158,97,153)(93,157,98,152)(94,156,99,151)(95,155,100,160)(111,146,116,141)(112,145,117,150)(113,144,118,149)(114,143,119,148)(115,142,120,147)(121,136,126,131)(122,135,127,140)(123,134,128,139)(124,133,129,138)(125,132,130,137), (1,133,51,97,24,145,62,110)(2,140,60,100,25,142,61,103)(3,137,59,93,26,149,70,106)(4,134,58,96,27,146,69,109)(5,131,57,99,28,143,68,102)(6,138,56,92,29,150,67,105)(7,135,55,95,30,147,66,108)(8,132,54,98,21,144,65,101)(9,139,53,91,22,141,64,104)(10,136,52,94,23,148,63,107)(11,90,121,44,151,77,119,33)(12,87,130,47,152,74,118,36)(13,84,129,50,153,71,117,39)(14,81,128,43,154,78,116,32)(15,88,127,46,155,75,115,35)(16,85,126,49,156,72,114,38)(17,82,125,42,157,79,113,31)(18,89,124,45,158,76,112,34)(19,86,123,48,159,73,111,37)(20,83,122,41,160,80,120,40), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,61)(11,100)(12,91)(13,92)(14,93)(15,94)(16,95)(17,96)(18,97)(19,98)(20,99)(21,58)(22,59)(23,60)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,81)(32,82)(33,83)(34,84)(35,85)(36,86)(37,87)(38,88)(39,89)(40,90)(41,77)(42,78)(43,79)(44,80)(45,71)(46,72)(47,73)(48,74)(49,75)(50,76)(101,159)(102,160)(103,151)(104,152)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,144)(112,145)(113,146)(114,147)(115,148)(116,149)(117,150)(118,141)(119,142)(120,143)(121,140)(122,131)(123,132)(124,133)(125,134)(126,135)(127,136)(128,137)(129,138)(130,139) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160)], [(1,89,6,84),(2,88,7,83),(3,87,8,82),(4,86,9,81),(5,85,10,90),(11,107,16,102),(12,106,17,101),(13,105,18,110),(14,104,19,109),(15,103,20,108),(21,79,26,74),(22,78,27,73),(23,77,28,72),(24,76,29,71),(25,75,30,80),(31,70,36,65),(32,69,37,64),(33,68,38,63),(34,67,39,62),(35,66,40,61),(41,60,46,55),(42,59,47,54),(43,58,48,53),(44,57,49,52),(45,56,50,51),(91,159,96,154),(92,158,97,153),(93,157,98,152),(94,156,99,151),(95,155,100,160),(111,146,116,141),(112,145,117,150),(113,144,118,149),(114,143,119,148),(115,142,120,147),(121,136,126,131),(122,135,127,140),(123,134,128,139),(124,133,129,138),(125,132,130,137)], [(1,133,51,97,24,145,62,110),(2,140,60,100,25,142,61,103),(3,137,59,93,26,149,70,106),(4,134,58,96,27,146,69,109),(5,131,57,99,28,143,68,102),(6,138,56,92,29,150,67,105),(7,135,55,95,30,147,66,108),(8,132,54,98,21,144,65,101),(9,139,53,91,22,141,64,104),(10,136,52,94,23,148,63,107),(11,90,121,44,151,77,119,33),(12,87,130,47,152,74,118,36),(13,84,129,50,153,71,117,39),(14,81,128,43,154,78,116,32),(15,88,127,46,155,75,115,35),(16,85,126,49,156,72,114,38),(17,82,125,42,157,79,113,31),(18,89,124,45,158,76,112,34),(19,86,123,48,159,73,111,37),(20,83,122,41,160,80,120,40)], [(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,61),(11,100),(12,91),(13,92),(14,93),(15,94),(16,95),(17,96),(18,97),(19,98),(20,99),(21,58),(22,59),(23,60),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,81),(32,82),(33,83),(34,84),(35,85),(36,86),(37,87),(38,88),(39,89),(40,90),(41,77),(42,78),(43,79),(44,80),(45,71),(46,72),(47,73),(48,74),(49,75),(50,76),(101,159),(102,160),(103,151),(104,152),(105,153),(106,154),(107,155),(108,156),(109,157),(110,158),(111,144),(112,145),(113,146),(114,147),(115,148),(116,149),(117,150),(118,141),(119,142),(120,143),(121,140),(122,131),(123,132),(124,133),(125,134),(126,135),(127,136),(128,137),(129,138),(130,139)]])

Matrix representation of Dic5.SD16 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 1 1 1 1
,
 0 40 0 0 0 0 1 0 0 0 0 0 0 0 27 13 22 32 0 0 27 36 5 14 0 0 9 19 28 14 0 0 10 19 5 32
,
 26 15 0 0 0 0 15 15 0 0 0 0 0 0 21 20 13 16 0 0 34 37 21 1 0 0 25 5 4 38 0 0 40 33 36 20
,
 40 0 0 0 0 0 0 1 0 0 0 0 0 0 36 31 9 22 0 0 19 14 9 28 0 0 13 32 27 22 0 0 19 32 10 5

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,40,0,0,1,0,0,0,40,0,1,0,0,0,0,40,1],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,27,27,9,10,0,0,13,36,19,19,0,0,22,5,28,5,0,0,32,14,14,32],[26,15,0,0,0,0,15,15,0,0,0,0,0,0,21,34,25,40,0,0,20,37,5,33,0,0,13,21,4,36,0,0,16,1,38,20],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,36,19,13,19,0,0,31,14,32,32,0,0,9,9,27,10,0,0,22,28,22,5] >;

Dic5.SD16 in GAP, Magma, Sage, TeX

{\rm Dic}_5.{\rm SD}_{16}
% in TeX

G:=Group("Dic5.SD16");
// GroupNames label

G:=SmallGroup(320,263);
// by ID

G=gap.SmallGroup(320,263);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,100,1571,570,136,6278,3156]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^8=d^2=1,b^2=a^5,b*a*b^-1=a^-1,c*a*c^-1=a^3,a*d=d*a,c*b*c^-1=d*b*d=a^5*b,d*c*d=b*c^3>;
// generators/relations

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