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G = C10.1442+ 1+4order 320 = 26·5

53rd non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C4○D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.1442+ 1+4, C10.1062- 1+4, C4○D45Dic5, D48(C2×Dic5), Q87(C2×Dic5), (D4×Dic5)⋊41C2, (Q8×Dic5)⋊28C2, (C2×D4).253D10, C10.72(C23×C4), (C2×Q8).209D10, C2.5(D48D10), (C2×C10).313C24, (C2×C20).560C23, C20.159(C22×C4), (C22×C4).286D10, C4.22(C22×Dic5), C2.13(C23×Dic5), C22.49(C23×D5), (D4×C10).275C22, C4⋊Dic5.392C22, (Q8×C10).242C22, C23.210(C22×D5), C2.5(D4.10D10), C23.21D1037C2, C22.4(C22×Dic5), (C22×C10).239C23, (C22×C20).295C22, C56(C23.33C23), (C2×Dic5).302C23, (C4×Dic5).183C22, C23.D5.135C22, (C22×Dic5).167C22, (C2×C20)⋊30(C2×C4), (C5×C4○D4)⋊12C4, (C5×D4)⋊32(C2×C4), (C5×Q8)⋊29(C2×C4), (C2×C4)⋊5(C2×Dic5), (C2×C4⋊Dic5)⋊47C2, (C2×C4○D4).12D5, (C10×C4○D4).14C2, (C2×C4).638(C22×D5), (C2×C10).132(C22×C4), SmallGroup(320,1499)

Series: Derived Chief Lower central Upper central

C1C10 — C10.1442+ 1+4
C1C5C10C2×C10C2×Dic5C22×Dic5D4×Dic5 — C10.1442+ 1+4
C5C10 — C10.1442+ 1+4
C1C22C2×C4○D4

Generators and relations for C10.1442+ 1+4
 G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, e2=a5, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 734 in 294 conjugacy classes, 191 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C4 [×8], C22, C22 [×6], C22 [×6], C5, C2×C4, C2×C4 [×15], C2×C4 [×14], D4 [×12], Q8 [×4], C23 [×3], C10 [×3], C10 [×6], C42 [×6], C22⋊C4 [×6], C4⋊C4 [×10], C22×C4 [×3], C22×C4 [×6], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×8], C20 [×8], C2×C10, C2×C10 [×6], C2×C10 [×6], C2×C4⋊C4 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C2×C4○D4, C2×Dic5 [×8], C2×Dic5 [×6], C2×C20, C2×C20 [×15], C5×D4 [×12], C5×Q8 [×4], C22×C10 [×3], C23.33C23, C4×Dic5 [×6], C4⋊Dic5, C4⋊Dic5 [×9], C23.D5 [×6], C22×Dic5 [×6], C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C2×C4⋊Dic5 [×3], C23.21D10 [×3], D4×Dic5 [×6], Q8×Dic5 [×2], C10×C4○D4, C10.1442+ 1+4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, Dic5 [×8], D10 [×7], C23×C4, 2+ 1+4, 2- 1+4, C2×Dic5 [×28], C22×D5 [×7], C23.33C23, C22×Dic5 [×14], C23×D5, D48D10, D4.10D10, C23×Dic5, C10.1442+ 1+4

Smallest permutation representation of C10.1442+ 1+4
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 78 30 81)(2 79 21 82)(3 80 22 83)(4 71 23 84)(5 72 24 85)(6 73 25 86)(7 74 26 87)(8 75 27 88)(9 76 28 89)(10 77 29 90)(11 106 153 93)(12 107 154 94)(13 108 155 95)(14 109 156 96)(15 110 157 97)(16 101 158 98)(17 102 159 99)(18 103 160 100)(19 104 151 91)(20 105 152 92)(31 63 43 51)(32 64 44 52)(33 65 45 53)(34 66 46 54)(35 67 47 55)(36 68 48 56)(37 69 49 57)(38 70 50 58)(39 61 41 59)(40 62 42 60)(111 131 124 144)(112 132 125 145)(113 133 126 146)(114 134 127 147)(115 135 128 148)(116 136 129 149)(117 137 130 150)(118 138 121 141)(119 139 122 142)(120 140 123 143)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 71)(10 72)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 87)(22 88)(23 89)(24 90)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 68)(32 69)(33 70)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 54)(42 55)(43 56)(44 57)(45 58)(46 59)(47 60)(48 51)(49 52)(50 53)(91 156)(92 157)(93 158)(94 159)(95 160)(96 151)(97 152)(98 153)(99 154)(100 155)(111 149)(112 150)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 133)(122 134)(123 135)(124 136)(125 137)(126 138)(127 139)(128 140)(129 131)(130 132)
(1 50 30 38)(2 41 21 39)(3 42 22 40)(4 43 23 31)(5 44 24 32)(6 45 25 33)(7 46 26 34)(8 47 27 35)(9 48 28 36)(10 49 29 37)(11 133 153 146)(12 134 154 147)(13 135 155 148)(14 136 156 149)(15 137 157 150)(16 138 158 141)(17 139 159 142)(18 140 160 143)(19 131 151 144)(20 132 152 145)(51 84 63 71)(52 85 64 72)(53 86 65 73)(54 87 66 74)(55 88 67 75)(56 89 68 76)(57 90 69 77)(58 81 70 78)(59 82 61 79)(60 83 62 80)(91 111 104 124)(92 112 105 125)(93 113 106 126)(94 114 107 127)(95 115 108 128)(96 116 109 129)(97 117 110 130)(98 118 101 121)(99 119 102 122)(100 120 103 123)
(1 132 6 137)(2 131 7 136)(3 140 8 135)(4 139 9 134)(5 138 10 133)(11 44 16 49)(12 43 17 48)(13 42 18 47)(14 41 19 46)(15 50 20 45)(21 144 26 149)(22 143 27 148)(23 142 28 147)(24 141 29 146)(25 150 30 145)(31 159 36 154)(32 158 37 153)(33 157 38 152)(34 156 39 151)(35 155 40 160)(51 102 56 107)(52 101 57 106)(53 110 58 105)(54 109 59 104)(55 108 60 103)(61 91 66 96)(62 100 67 95)(63 99 68 94)(64 98 69 93)(65 97 70 92)(71 122 76 127)(72 121 77 126)(73 130 78 125)(74 129 79 124)(75 128 80 123)(81 112 86 117)(82 111 87 116)(83 120 88 115)(84 119 89 114)(85 118 90 113)

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

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

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

74 conjugacy classes

class 1 2A2B2C2D···2I4A···4H4I···4X5A5B10A···10F10G···10R20A···20H20I···20T
order12222···24···44···45510···1010···1020···2020···20
size11112···22···210···10222···24···42···24···4

74 irreducible representations

dim1111111222224444
type++++++++++-+-+-
imageC1C2C2C2C2C2C4D5D10D10D10Dic52+ 1+42- 1+4D48D10D4.10D10
kernelC10.1442+ 1+4C2×C4⋊Dic5C23.21D10D4×Dic5Q8×Dic5C10×C4○D4C5×C4○D4C2×C4○D4C22×C4C2×D4C2×Q8C4○D4C10C10C2C2
# reps133621162662161144

Matrix representation of C10.1442+ 1+4 in GL6(𝔽41)

0340000
6350000
0040700
0034700
0061367
00822350
,
100000
010000
00951821
001138021
001227234
0032341433
,
100000
010000
0032362320
00303020
00385397
0093278
,
4000000
0400000
00113200
0093000
00436232
001913739
,
2310000
3180000
00141400
00302700
0011262514
0022331416

G:=sub<GL(6,GF(41))| [0,6,0,0,0,0,34,35,0,0,0,0,0,0,40,34,6,8,0,0,7,7,13,22,0,0,0,0,6,35,0,0,0,0,7,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,11,12,32,0,0,5,38,27,34,0,0,18,0,2,14,0,0,21,21,34,33],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,30,38,9,0,0,36,3,5,3,0,0,23,0,39,27,0,0,20,20,7,8],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,11,9,4,19,0,0,32,30,36,1,0,0,0,0,2,37,0,0,0,0,32,39],[23,3,0,0,0,0,1,18,0,0,0,0,0,0,14,30,11,22,0,0,14,27,26,33,0,0,0,0,25,14,0,0,0,0,14,16] >;

C10.1442+ 1+4 in GAP, Magma, Sage, TeX

C_{10}._{144}2_+^{1+4}
% in TeX

G:=Group("C10.144ES+(2,2)");
// GroupNames label

G:=SmallGroup(320,1499);
// by ID

G=gap.SmallGroup(320,1499);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=b^2,e^2=a^5,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;
// generators/relations

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