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G = (C5×D4)⋊14D4order 320 = 26·5

2nd semidirect product of C5×D4 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C5×D4)⋊14D4, (C5×Q8)⋊14D4, C57(D4⋊D4), D46(C5⋊D4), Q86(C5⋊D4), C207D426C2, C20.215(C2×D4), (C2×C20).452D4, C10.78C22≀C2, (C2×D4).204D10, D4⋊Dic541C2, Q8⋊Dic541C2, (C2×Q8).173D10, C10.112(C4○D8), C20.55D418C2, (C2×C20).484C23, (C22×C4).164D10, (C22×C10).115D4, C23.32(C5⋊D4), C2.23(D4⋊D10), C10.125(C8⋊C22), (C2×D20).137C22, (D4×C10).245C22, C2.12(C242D5), C4⋊Dic5.189C22, (Q8×C10).208C22, C2.30(D4.8D10), (C22×C20).210C22, (C2×C4○D4)⋊1D5, (C2×D4⋊D5)⋊24C2, (C10×C4○D4)⋊1C2, (C2×Q8⋊D5)⋊24C2, C4.62(C2×C5⋊D4), (C2×C10).567(C2×D4), (C2×C4).225(C5⋊D4), (C2×C4).568(C22×D5), C22.224(C2×C5⋊D4), (C2×C52C8).178C22, SmallGroup(320,865)

Series: Derived Chief Lower central Upper central

C1C2×C20 — (C5×D4)⋊14D4
C1C5C10C2×C10C2×C20C2×D20C2×D4⋊D5 — (C5×D4)⋊14D4
C5C10C2×C20 — (C5×D4)⋊14D4
C1C22C22×C4C2×C4○D4

Generators and relations for (C5×D4)⋊14D4
 G = < a,b,c,d,e | a5=b4=c2=d4=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, cbc=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=d-1 >

Subgroups: 590 in 162 conjugacy classes, 47 normal (39 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×10], C5, C8 [×2], C2×C4 [×2], C2×C4 [×8], D4 [×2], D4 [×9], Q8 [×2], Q8, C23, C23 [×2], D5, C10 [×3], C10 [×3], C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×2], SD16 [×2], C22×C4, C22×C4, C2×D4, C2×D4 [×3], C2×Q8, C4○D4 [×4], Dic5, C20 [×2], C20 [×3], D10 [×3], C2×C10, C2×C10 [×7], C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C52C8 [×2], D20 [×2], C2×Dic5, C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×7], C5×D4 [×2], C5×D4 [×5], C5×Q8 [×2], C5×Q8, C22×D5, C22×C10, C22×C10, D4⋊D4, C2×C52C8 [×2], C4⋊Dic5, D10⋊C4, D4⋊D5 [×2], Q8⋊D5 [×2], C2×D20, C2×C5⋊D4, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4 [×4], C20.55D4, D4⋊Dic5, Q8⋊Dic5, C207D4, C2×D4⋊D5, C2×Q8⋊D5, C10×C4○D4, (C5×D4)⋊14D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C4○D8, C8⋊C22, C5⋊D4 [×6], C22×D5, D4⋊D4, C2×C5⋊D4 [×3], D4⋊D10, D4.8D10, C242D5, (C5×D4)⋊14D4

Smallest permutation representation of (C5×D4)⋊14D4
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 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 75)(81 91 86 96)(82 92 87 97)(83 93 88 98)(84 94 89 99)(85 95 90 100)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)(121 131 126 136)(122 132 127 137)(123 133 128 138)(124 134 129 139)(125 135 130 140)(141 151 146 156)(142 152 147 157)(143 153 148 158)(144 154 149 159)(145 155 150 160)
(1 116)(2 117)(3 118)(4 119)(5 120)(6 111)(7 112)(8 113)(9 114)(10 115)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 96)(22 97)(23 98)(24 99)(25 100)(26 91)(27 92)(28 93)(29 94)(30 95)(31 81)(32 82)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 156)(42 157)(43 158)(44 159)(45 160)(46 151)(47 152)(48 153)(49 154)(50 155)(51 141)(52 142)(53 143)(54 144)(55 145)(56 146)(57 147)(58 148)(59 149)(60 150)(61 136)(62 137)(63 138)(64 139)(65 140)(66 131)(67 132)(68 133)(69 134)(70 135)(71 121)(72 122)(73 123)(74 124)(75 125)(76 126)(77 127)(78 128)(79 129)(80 130)
(1 51 26 76)(2 55 27 80)(3 54 28 79)(4 53 29 78)(5 52 30 77)(6 56 21 71)(7 60 22 75)(8 59 23 74)(9 58 24 73)(10 57 25 72)(11 41 36 66)(12 45 37 70)(13 44 38 69)(14 43 39 68)(15 42 40 67)(16 46 31 61)(17 50 32 65)(18 49 33 64)(19 48 34 63)(20 47 35 62)(81 121 106 146)(82 125 107 150)(83 124 108 149)(84 123 109 148)(85 122 110 147)(86 126 101 141)(87 130 102 145)(88 129 103 144)(89 128 104 143)(90 127 105 142)(91 136 116 151)(92 140 117 155)(93 139 118 154)(94 138 119 153)(95 137 120 152)(96 131 111 156)(97 135 112 160)(98 134 113 159)(99 133 114 158)(100 132 115 157)
(2 5)(3 4)(7 10)(8 9)(11 16)(12 20)(13 19)(14 18)(15 17)(22 25)(23 24)(27 30)(28 29)(31 36)(32 40)(33 39)(34 38)(35 37)(41 61)(42 65)(43 64)(44 63)(45 62)(46 66)(47 70)(48 69)(49 68)(50 67)(51 76)(52 80)(53 79)(54 78)(55 77)(56 71)(57 75)(58 74)(59 73)(60 72)(81 91)(82 95)(83 94)(84 93)(85 92)(86 96)(87 100)(88 99)(89 98)(90 97)(101 111)(102 115)(103 114)(104 113)(105 112)(106 116)(107 120)(108 119)(109 118)(110 117)(121 151)(122 155)(123 154)(124 153)(125 152)(126 156)(127 160)(128 159)(129 158)(130 157)(131 141)(132 145)(133 144)(134 143)(135 142)(136 146)(137 150)(138 149)(139 148)(140 147)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G5A5B8A8B8C8D10A···10F10G···10R20A···20H20I···20T
order12222222444444455888810···1010···1020···2020···20
size1111444402222444022202020202···24···42···24···4

59 irreducible representations

dim111111112222222222222444
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D4D5D10D10D10C4○D8C5⋊D4C5⋊D4C5⋊D4C5⋊D4C8⋊C22D4⋊D10D4.8D10
kernel(C5×D4)⋊14D4C20.55D4D4⋊Dic5Q8⋊Dic5C207D4C2×D4⋊D5C2×Q8⋊D5C10×C4○D4C2×C20C5×D4C5×Q8C22×C10C2×C4○D4C22×C4C2×D4C2×Q8C10C2×C4D4Q8C23C10C2C2
# reps111111111221222244884144

Matrix representation of (C5×D4)⋊14D4 in GL4(𝔽41) generated by

64000
1000
0010
0001
,
1000
0100
0014
002040
,
1000
0100
001734
00624
,
183500
202300
0090
001632
,
1000
64000
0014
00040
G:=sub<GL(4,GF(41))| [6,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,20,0,0,4,40],[1,0,0,0,0,1,0,0,0,0,17,6,0,0,34,24],[18,20,0,0,35,23,0,0,0,0,9,16,0,0,0,32],[1,6,0,0,0,40,0,0,0,0,1,0,0,0,4,40] >;

(C5×D4)⋊14D4 in GAP, Magma, Sage, TeX

(C_5\times D_4)\rtimes_{14}D_4
% in TeX

G:=Group("(C5xD4):14D4");
// GroupNames label

G:=SmallGroup(320,865);
// by ID

G=gap.SmallGroup(320,865);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,184,1684,851,102,12550]);
// Polycyclic

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

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