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G = (C2×Q16)⋊D5order 320 = 26·5

2nd semidirect product of C2×Q16 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×Q16)⋊2D5, (C5×Q8).8D4, (Q8×Dic5)⋊7C2, (C2×C8).37D10, (C10×Q16)⋊12C2, C20.185(C2×D4), (C2×Q8).61D10, Q8.3(C5⋊D4), C57(Q8.D4), C10.79(C4○D8), Q8⋊Dic532C2, C20.8Q829C2, (C2×Dic5).83D4, C22.276(D4×D5), D205C4.12C2, C20.106(C4○D4), C4.14(D42D5), (C2×C20).459C23, (C2×C40).251C22, C20.23D4.7C2, (Q8×C10).88C22, C2.16(Q8.D10), C10.120(C4⋊D4), C2.27(Q16⋊D5), (C2×D20).128C22, C10.77(C8.C22), C4⋊Dic5.182C22, (C4×Dic5).61C22, C2.29(Dic5⋊D4), C4.47(C2×C5⋊D4), (C2×Q8⋊D5).8C2, (C2×C10).370(C2×D4), (C2×C4).547(C22×D5), (C2×C52C8).164C22, SmallGroup(320,812)

Series: Derived Chief Lower central Upper central

C1C2×C20 — (C2×Q16)⋊D5
C1C5C10C20C2×C20C4×Dic5Q8×Dic5 — (C2×Q16)⋊D5
C5C10C2×C20 — (C2×Q16)⋊D5
C1C22C2×C4C2×Q16

Generators and relations for (C2×Q16)⋊D5
 G = < a,b,c,d,e | a2=b8=d5=e2=1, c2=b4, ab=ba, ece=ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=ab-1, cd=dc, ede=d-1 >

Subgroups: 454 in 112 conjugacy classes, 41 normal (37 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×6], C22, C22 [×3], C5, C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], Q8 [×2], Q8 [×3], C23, D5, C10 [×3], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8, SD16 [×2], Q16 [×2], C2×D4, C2×Q8 [×2], Dic5 [×3], C20 [×2], C20 [×3], D10 [×3], C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C52C8, C40, D20 [×2], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C5×Q8 [×3], C22×D5, Q8.D4, C2×C52C8, C4×Dic5, C4×Dic5, C4⋊Dic5, C4⋊Dic5, D10⋊C4 [×2], Q8⋊D5 [×2], C2×C40, C5×Q16 [×2], C2×D20, Q8×C10 [×2], C20.8Q8, D205C4, Q8⋊Dic5, C2×Q8⋊D5, Q8×Dic5, C20.23D4, C10×Q16, (C2×Q16)⋊D5
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C4○D8, C8.C22, C5⋊D4 [×2], C22×D5, Q8.D4, D4×D5, D42D5, C2×C5⋊D4, Q16⋊D5, Q8.D10, Dic5⋊D4, (C2×Q16)⋊D5

Smallest permutation representation of (C2×Q16)⋊D5
On 160 points
Generators in S160
(1 147)(2 148)(3 149)(4 150)(5 151)(6 152)(7 145)(8 146)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 57)(17 106)(18 107)(19 108)(20 109)(21 110)(22 111)(23 112)(24 105)(25 86)(26 87)(27 88)(28 81)(29 82)(30 83)(31 84)(32 85)(33 76)(34 77)(35 78)(36 79)(37 80)(38 73)(39 74)(40 75)(41 127)(42 128)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 154)(50 155)(51 156)(52 157)(53 158)(54 159)(55 160)(56 153)(65 140)(66 141)(67 142)(68 143)(69 144)(70 137)(71 138)(72 139)(89 117)(90 118)(91 119)(92 120)(93 113)(94 114)(95 115)(96 116)(97 134)(98 135)(99 136)(100 129)(101 130)(102 131)(103 132)(104 133)
(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 12 5 16)(2 11 6 15)(3 10 7 14)(4 9 8 13)(17 77 21 73)(18 76 22 80)(19 75 23 79)(20 74 24 78)(25 91 29 95)(26 90 30 94)(27 89 31 93)(28 96 32 92)(33 111 37 107)(34 110 38 106)(35 109 39 105)(36 108 40 112)(41 132 45 136)(42 131 46 135)(43 130 47 134)(44 129 48 133)(49 141 53 137)(50 140 54 144)(51 139 55 143)(52 138 56 142)(57 147 61 151)(58 146 62 150)(59 145 63 149)(60 152 64 148)(65 159 69 155)(66 158 70 154)(67 157 71 153)(68 156 72 160)(81 116 85 120)(82 115 86 119)(83 114 87 118)(84 113 88 117)(97 121 101 125)(98 128 102 124)(99 127 103 123)(100 126 104 122)
(1 65 125 31 105)(2 66 126 32 106)(3 67 127 25 107)(4 68 128 26 108)(5 69 121 27 109)(6 70 122 28 110)(7 71 123 29 111)(8 72 124 30 112)(9 156 102 90 40)(10 157 103 91 33)(11 158 104 92 34)(12 159 97 93 35)(13 160 98 94 36)(14 153 99 95 37)(15 154 100 96 38)(16 155 101 89 39)(17 148 141 48 85)(18 149 142 41 86)(19 150 143 42 87)(20 151 144 43 88)(21 152 137 44 81)(22 145 138 45 82)(23 146 139 46 83)(24 147 140 47 84)(49 129 116 73 64)(50 130 117 74 57)(51 131 118 75 58)(52 132 119 76 59)(53 133 120 77 60)(54 134 113 78 61)(55 135 114 79 62)(56 136 115 80 63)
(1 105)(2 23)(3 111)(4 21)(5 109)(6 19)(7 107)(8 17)(9 38)(10 80)(11 36)(12 78)(13 34)(14 76)(15 40)(16 74)(18 145)(20 151)(22 149)(24 147)(25 71)(26 137)(27 69)(28 143)(29 67)(30 141)(31 65)(32 139)(33 63)(35 61)(37 59)(39 57)(41 45)(42 122)(44 128)(46 126)(48 124)(49 118)(50 89)(51 116)(52 95)(53 114)(54 93)(55 120)(56 91)(58 73)(60 79)(62 77)(64 75)(66 83)(68 81)(70 87)(72 85)(82 142)(84 140)(86 138)(88 144)(90 154)(92 160)(94 158)(96 156)(97 134)(98 104)(99 132)(100 102)(101 130)(103 136)(106 146)(108 152)(110 150)(112 148)(113 159)(115 157)(117 155)(119 153)(123 127)(129 131)(133 135)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222444444444455888810···102020202020···2040···40
size111140224481010202020224420202···244448···84···4

47 irreducible representations

dim111111112222222244444
type+++++++++++++--++
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4D10D10C4○D8C5⋊D4C8.C22D42D5D4×D5Q16⋊D5Q8.D10
kernel(C2×Q16)⋊D5C20.8Q8D205C4Q8⋊Dic5C2×Q8⋊D5Q8×Dic5C20.23D4C10×Q16C2×Dic5C5×Q8C2×Q16C20C2×C8C2×Q8C10Q8C10C4C22C2C2
# reps111111112222244812244

Matrix representation of (C2×Q16)⋊D5 in GL6(𝔽41)

4000000
0400000
001000
000100
000010
000001
,
900000
9320000
0040000
0004000
0000015
00003024
,
4020000
010000
0040000
0004000
0000320
000029
,
100000
010000
0040100
0033700
000010
000001
,
100000
1400000
0040000
0033100
000010
00001840

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,9,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,30,0,0,0,0,15,24],[40,0,0,0,0,0,2,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,2,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,33,0,0,0,0,1,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,40,33,0,0,0,0,0,1,0,0,0,0,0,0,1,18,0,0,0,0,0,40] >;

(C2×Q16)⋊D5 in GAP, Magma, Sage, TeX

(C_2\times Q_{16})\rtimes D_5
% in TeX

G:=Group("(C2xQ16):D5");
// GroupNames label

G:=SmallGroup(320,812);
// by ID

G=gap.SmallGroup(320,812);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,1094,135,184,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^8=d^5=e^2=1,c^2=b^4,a*b=b*a,e*c*e=a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,e*b*e=a*b^-1,c*d=d*c,e*d*e=d^-1>;
// generators/relations

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