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G = C5×Q85D4order 320 = 26·5

Direct product of C5 and Q85D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×Q85D4, C10.1182- 1+4, Q85(C5×D4), (C5×Q8)⋊23D4, (C4×D4)⋊17C10, (D4×C20)⋊46C2, (Q8×C20)⋊31C2, (C4×Q8)⋊11C10, C4.42(D4×C10), C4⋊D413C10, C20.403(C2×D4), C22⋊Q813C10, (C22×Q8)⋊6C10, C4.4D411C10, C42.43(C2×C10), (C2×C10).368C24, (C4×C20).284C22, (C2×C20).714C23, C10.196(C22×D4), (D4×C10).322C22, C22.42(C23×C10), C23.42(C22×C10), (Q8×C10).274C22, C2.10(C5×2- 1+4), (C22×C10).100C23, (C22×C20).454C22, (Q8×C2×C10)⋊18C2, C2.20(D4×C2×C10), (C2×C4○D4)⋊8C10, C223(C5×C4○D4), (C10×C4○D4)⋊24C2, (C5×C4⋊D4)⋊40C2, C4⋊C4.71(C2×C10), C2.22(C10×C4○D4), (C5×C22⋊Q8)⋊40C2, (C2×C10)⋊17(C4○D4), (C2×D4).67(C2×C10), C10.241(C2×C4○D4), (C5×C4.4D4)⋊31C2, (C2×Q8).61(C2×C10), C22⋊C4.19(C2×C10), (C5×C4⋊C4).396C22, (C22×C4).66(C2×C10), (C2×C4).60(C22×C10), (C5×C22⋊C4).88C22, SmallGroup(320,1550)

Series: Derived Chief Lower central Upper central

C1C22 — C5×Q85D4
C1C2C22C2×C10C22×C10D4×C10C5×C4⋊D4 — C5×Q85D4
C1C22 — C5×Q85D4
C1C2×C10 — C5×Q85D4

Generators and relations for C5×Q85D4
 G = < a,b,c,d,e | a5=b4=d4=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=ece=b2c, ede=d-1 >

Subgroups: 426 in 290 conjugacy classes, 166 normal (28 characteristic)
C1, C2 [×3], C2 [×5], C4 [×6], C4 [×8], C22, C22 [×2], C22 [×11], C5, C2×C4 [×2], C2×C4 [×9], C2×C4 [×12], D4 [×12], Q8 [×4], Q8 [×6], C23, C23 [×3], C10 [×3], C10 [×5], C42 [×3], C22⋊C4, C22⋊C4 [×9], C4⋊C4 [×6], C22×C4 [×6], C2×D4 [×6], C2×Q8, C2×Q8 [×3], C2×Q8 [×4], C4○D4 [×4], C20 [×6], C20 [×8], C2×C10, C2×C10 [×2], C2×C10 [×11], C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4 [×3], C22×Q8, C2×C4○D4, C2×C20 [×2], C2×C20 [×9], C2×C20 [×12], C5×D4 [×12], C5×Q8 [×4], C5×Q8 [×6], C22×C10, C22×C10 [×3], Q85D4, C4×C20 [×3], C5×C22⋊C4, C5×C22⋊C4 [×9], C5×C4⋊C4 [×6], C22×C20 [×6], D4×C10 [×6], Q8×C10, Q8×C10 [×3], Q8×C10 [×4], C5×C4○D4 [×4], D4×C20 [×3], Q8×C20, C5×C4⋊D4 [×3], C5×C22⋊Q8 [×3], C5×C4.4D4 [×3], Q8×C2×C10, C10×C4○D4, C5×Q85D4
Quotients: C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C4○D4 [×2], C24, C2×C10 [×35], C22×D4, C2×C4○D4, 2- 1+4, C5×D4 [×4], C22×C10 [×15], Q85D4, D4×C10 [×6], C5×C4○D4 [×2], C23×C10, D4×C2×C10, C10×C4○D4, C5×2- 1+4, C5×Q85D4

Smallest permutation representation of C5×Q85D4
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 47 35 41)(2 48 31 42)(3 49 32 43)(4 50 33 44)(5 46 34 45)(6 145 160 146)(7 141 156 147)(8 142 157 148)(9 143 158 149)(10 144 159 150)(11 151 17 137)(12 152 18 138)(13 153 19 139)(14 154 20 140)(15 155 16 136)(21 55 27 36)(22 51 28 37)(23 52 29 38)(24 53 30 39)(25 54 26 40)(56 87 75 81)(57 88 71 82)(58 89 72 83)(59 90 73 84)(60 86 74 85)(61 95 67 76)(62 91 68 77)(63 92 69 78)(64 93 70 79)(65 94 66 80)(96 121 115 127)(97 122 111 128)(98 123 112 129)(99 124 113 130)(100 125 114 126)(101 116 107 135)(102 117 108 131)(103 118 109 132)(104 119 110 133)(105 120 106 134)
(1 107 35 101)(2 108 31 102)(3 109 32 103)(4 110 33 104)(5 106 34 105)(6 86 160 85)(7 87 156 81)(8 88 157 82)(9 89 158 83)(10 90 159 84)(11 77 17 91)(12 78 18 92)(13 79 19 93)(14 80 20 94)(15 76 16 95)(21 115 27 96)(22 111 28 97)(23 112 29 98)(24 113 30 99)(25 114 26 100)(36 127 55 121)(37 128 51 122)(38 129 52 123)(39 130 53 124)(40 126 54 125)(41 135 47 116)(42 131 48 117)(43 132 49 118)(44 133 50 119)(45 134 46 120)(56 147 75 141)(57 148 71 142)(58 149 72 143)(59 150 73 144)(60 146 74 145)(61 155 67 136)(62 151 68 137)(63 152 69 138)(64 153 70 139)(65 154 66 140)
(1 136 21 141)(2 137 22 142)(3 138 23 143)(4 139 24 144)(5 140 25 145)(6 45 20 40)(7 41 16 36)(8 42 17 37)(9 43 18 38)(10 44 19 39)(11 51 157 48)(12 52 158 49)(13 53 159 50)(14 54 160 46)(15 55 156 47)(26 146 34 154)(27 147 35 155)(28 148 31 151)(29 149 32 152)(30 150 33 153)(56 101 61 96)(57 102 62 97)(58 103 63 98)(59 104 64 99)(60 105 65 100)(66 114 74 106)(67 115 75 107)(68 111 71 108)(69 112 72 109)(70 113 73 110)(76 127 81 135)(77 128 82 131)(78 129 83 132)(79 130 84 133)(80 126 85 134)(86 120 94 125)(87 116 95 121)(88 117 91 122)(89 118 92 123)(90 119 93 124)
(1 96)(2 97)(3 98)(4 99)(5 100)(6 85)(7 81)(8 82)(9 83)(10 84)(11 91)(12 92)(13 93)(14 94)(15 95)(16 76)(17 77)(18 78)(19 79)(20 80)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 135)(37 131)(38 132)(39 133)(40 134)(41 127)(42 128)(43 129)(44 130)(45 126)(46 125)(47 121)(48 122)(49 123)(50 124)(51 117)(52 118)(53 119)(54 120)(55 116)(56 141)(57 142)(58 143)(59 144)(60 145)(61 136)(62 137)(63 138)(64 139)(65 140)(66 154)(67 155)(68 151)(69 152)(70 153)(71 148)(72 149)(73 150)(74 146)(75 147)(86 160)(87 156)(88 157)(89 158)(90 159)

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

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

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

125 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4J4K···4P5A5B5C5D10A···10L10M···10T10U···10AF20A···20AN20AO···20BL
order1222222224···44···4555510···1010···1010···1020···2020···20
size1111224442···24···411111···12···24···42···24···4

125 irreducible representations

dim1111111111111111222244
type+++++++++-
imageC1C2C2C2C2C2C2C2C5C10C10C10C10C10C10C10D4C4○D4C5×D4C5×C4○D42- 1+4C5×2- 1+4
kernelC5×Q85D4D4×C20Q8×C20C5×C4⋊D4C5×C22⋊Q8C5×C4.4D4Q8×C2×C10C10×C4○D4Q85D4C4×D4C4×Q8C4⋊D4C22⋊Q8C4.4D4C22×Q8C2×C4○D4C5×Q8C2×C10Q8C22C10C2
# reps1313331141241212124444161614

Matrix representation of C5×Q85D4 in GL4(𝔽41) generated by

1000
0100
00160
00016
,
1000
0100
002939
001112
,
40000
04000
001523
00826
,
40200
40100
001523
001726
,
1000
14000
001523
001726
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,29,11,0,0,39,12],[40,0,0,0,0,40,0,0,0,0,15,8,0,0,23,26],[40,40,0,0,2,1,0,0,0,0,15,17,0,0,23,26],[1,1,0,0,0,40,0,0,0,0,15,17,0,0,23,26] >;

C5×Q85D4 in GAP, Magma, Sage, TeX

C_5\times Q_8\rtimes_5D_4
% in TeX

G:=Group("C5xQ8:5D4");
// GroupNames label

G:=SmallGroup(320,1550);
// by ID

G=gap.SmallGroup(320,1550);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,568,3446,1242,304]);
// Polycyclic

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

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