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## G = C2×C4×C5⋊D4order 320 = 26·5

### Direct product of C2×C4 and C5⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C4×C5⋊D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C22×C5⋊D4 — C2×C4×C5⋊D4
 Lower central C5 — C10 — C2×C4×C5⋊D4
 Upper central C1 — C22×C4 — C23×C4

Generators and relations for C2×C4×C5⋊D4
G = < a,b,c,d,e | a2=b4=c5=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1262 in 426 conjugacy classes, 183 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×4], C4 [×10], C22, C22 [×10], C22 [×28], C5, C2×C4 [×8], C2×C4 [×32], D4 [×16], C23, C23 [×6], C23 [×14], D5 [×4], C10 [×3], C10 [×4], C10 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×15], C2×D4 [×12], C24, C24, Dic5 [×4], Dic5 [×4], C20 [×4], C20 [×2], D10 [×4], D10 [×12], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C23×C4, C23×C4, C22×D4, C4×D5 [×8], C2×Dic5 [×10], C2×Dic5 [×4], C5⋊D4 [×16], C2×C20 [×8], C2×C20 [×10], C22×D5 [×6], C22×D5 [×4], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C4×D4, C4×Dic5 [×4], C10.D4 [×4], D10⋊C4 [×4], C23.D5 [×4], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5 [×3], C2×C5⋊D4 [×12], C22×C20 [×2], C22×C20 [×4], C22×C20 [×4], C23×D5, C23×C10, C2×C4×Dic5, C2×C10.D4, C2×D10⋊C4, C4×C5⋊D4 [×8], C2×C23.D5, D5×C22×C4, C22×C5⋊D4, C23×C20, C2×C4×C5⋊D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], D5, C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C4×D5 [×4], C5⋊D4 [×4], C22×D5 [×7], C2×C4×D4, C2×C4×D5 [×6], C4○D20 [×2], C2×C5⋊D4 [×6], C23×D5, C4×C5⋊D4 [×4], D5×C22×C4, C2×C4○D20, C22×C5⋊D4, C2×C4×C5⋊D4

Smallest permutation representation of C2×C4×C5⋊D4
On 160 points
Generators in S160
(1 86)(2 87)(3 88)(4 89)(5 90)(6 81)(7 82)(8 83)(9 84)(10 85)(11 96)(12 97)(13 98)(14 99)(15 100)(16 91)(17 92)(18 93)(19 94)(20 95)(21 106)(22 107)(23 108)(24 109)(25 110)(26 101)(27 102)(28 103)(29 104)(30 105)(31 116)(32 117)(33 118)(34 119)(35 120)(36 111)(37 112)(38 113)(39 114)(40 115)(41 126)(42 127)(43 128)(44 129)(45 130)(46 121)(47 122)(48 123)(49 124)(50 125)(51 136)(52 137)(53 138)(54 139)(55 140)(56 131)(57 132)(58 133)(59 134)(60 135)(61 146)(62 147)(63 148)(64 149)(65 150)(66 141)(67 142)(68 143)(69 144)(70 145)(71 156)(72 157)(73 158)(74 159)(75 160)(76 151)(77 152)(78 153)(79 154)(80 155)
(1 36 11 26)(2 37 12 27)(3 38 13 28)(4 39 14 29)(5 40 15 30)(6 31 16 21)(7 32 17 22)(8 33 18 23)(9 34 19 24)(10 35 20 25)(41 76 51 66)(42 77 52 67)(43 78 53 68)(44 79 54 69)(45 80 55 70)(46 71 56 61)(47 72 57 62)(48 73 58 63)(49 74 59 64)(50 75 60 65)(81 116 91 106)(82 117 92 107)(83 118 93 108)(84 119 94 109)(85 120 95 110)(86 111 96 101)(87 112 97 102)(88 113 98 103)(89 114 99 104)(90 115 100 105)(121 156 131 146)(122 157 132 147)(123 158 133 148)(124 159 134 149)(125 160 135 150)(126 151 136 141)(127 152 137 142)(128 153 138 143)(129 154 139 144)(130 155 140 145)
(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)
(1 121 6 126)(2 125 7 130)(3 124 8 129)(4 123 9 128)(5 122 10 127)(11 131 16 136)(12 135 17 140)(13 134 18 139)(14 133 19 138)(15 132 20 137)(21 141 26 146)(22 145 27 150)(23 144 28 149)(24 143 29 148)(25 142 30 147)(31 151 36 156)(32 155 37 160)(33 154 38 159)(34 153 39 158)(35 152 40 157)(41 86 46 81)(42 90 47 85)(43 89 48 84)(44 88 49 83)(45 87 50 82)(51 96 56 91)(52 100 57 95)(53 99 58 94)(54 98 59 93)(55 97 60 92)(61 106 66 101)(62 110 67 105)(63 109 68 104)(64 108 69 103)(65 107 70 102)(71 116 76 111)(72 120 77 115)(73 119 78 114)(74 118 79 113)(75 117 80 112)
(2 5)(3 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(41 46)(42 50)(43 49)(44 48)(45 47)(51 56)(52 60)(53 59)(54 58)(55 57)(61 66)(62 70)(63 69)(64 68)(65 67)(71 76)(72 80)(73 79)(74 78)(75 77)(82 85)(83 84)(87 90)(88 89)(92 95)(93 94)(97 100)(98 99)(102 105)(103 104)(107 110)(108 109)(112 115)(113 114)(117 120)(118 119)(121 126)(122 130)(123 129)(124 128)(125 127)(131 136)(132 140)(133 139)(134 138)(135 137)(141 146)(142 150)(143 149)(144 148)(145 147)(151 156)(152 160)(153 159)(154 158)(155 157)

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

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

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

104 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A ··· 4H 4I 4J 4K 4L 4M ··· 4X 5A 5B 10A ··· 10AD 20A ··· 20AF order 1 2 ··· 2 2 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 10 10 10 10 1 ··· 1 2 2 2 2 10 ··· 10 2 2 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 D4 D5 C4○D4 D10 D10 C5⋊D4 C4×D5 C4○D20 kernel C2×C4×C5⋊D4 C2×C4×Dic5 C2×C10.D4 C2×D10⋊C4 C4×C5⋊D4 C2×C23.D5 D5×C22×C4 C22×C5⋊D4 C23×C20 C2×C5⋊D4 C2×C20 C23×C4 C2×C10 C22×C4 C24 C2×C4 C23 C22 # reps 1 1 1 1 8 1 1 1 1 16 4 2 4 12 2 16 16 16

Matrix representation of C2×C4×C5⋊D4 in GL5(𝔽41)

 40 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 40
,
 40 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9 0 0 0 0 0 9
,
 1 0 0 0 0 0 0 40 0 0 0 1 34 0 0 0 0 0 40 40 0 0 0 8 7
,
 1 0 0 0 0 0 40 7 0 0 0 0 1 0 0 0 0 0 3 21 0 0 0 21 38
,
 40 0 0 0 0 0 1 34 0 0 0 0 40 0 0 0 0 0 34 35 0 0 0 8 7

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,40,34,0,0,0,0,0,40,8,0,0,0,40,7],[1,0,0,0,0,0,40,0,0,0,0,7,1,0,0,0,0,0,3,21,0,0,0,21,38],[40,0,0,0,0,0,1,0,0,0,0,34,40,0,0,0,0,0,34,8,0,0,0,35,7] >;

C2×C4×C5⋊D4 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_5\rtimes D_4
% in TeX

G:=Group("C2xC4xC5:D4");
// GroupNames label

G:=SmallGroup(320,1460);
// by ID

G=gap.SmallGroup(320,1460);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,80,12550]);
// Polycyclic

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

׿
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