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G = (C2×Q8).5F5order 320 = 26·5

2nd non-split extension by C2×Q8 of F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×Q8).5F5, (C2×D20).9C4, (C4×D5).45D4, (Q8×C10).6C4, D10⋊C84C2, C2.8(Q8.F5), C10.23(C8○D4), C4.21(C22⋊F5), C20.21(C22⋊C4), Dic5.113(C2×D4), D10.16(C22⋊C4), C22.97(C22×F5), (C2×Dic5).357C23, (C2×D5⋊C8)⋊3C2, (C2×C4.F5)⋊4C2, (C2×C4).41(C2×F5), (C2×C20).26(C2×C4), (C2×C5⋊C8).13C22, (C2×C4×D5).62C22, C2.31(C2×C22⋊F5), C53((C22×C8)⋊C2), C10.30(C2×C22⋊C4), (C2×C10).86(C22×C4), (C2×Q82D5).12C2, (C22×D5).58(C2×C4), SmallGroup(320,1125)

Series: Derived Chief Lower central Upper central

C1C2×C10 — (C2×Q8).5F5
C1C5C10Dic5C2×Dic5C2×C5⋊C8C2×D5⋊C8 — (C2×Q8).5F5
C5C2×C10 — (C2×Q8).5F5
C1C22C2×Q8

Generators and relations for (C2×Q8).5F5
 G = < a,b,c,d,e | a2=b4=d5=1, c2=e4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe-1=ab-1, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 650 in 158 conjugacy classes, 52 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×10], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×9], D4 [×6], Q8 [×2], C23 [×3], D5 [×4], C10, C10 [×2], C2×C8 [×6], M4(2) [×2], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×4], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×8], C2×C10, C22⋊C8 [×4], C22×C8, C2×M4(2), C2×C4○D4, C5⋊C8 [×4], C4×D5 [×4], C4×D5 [×4], D20 [×6], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, C22×D5 [×2], (C22×C8)⋊C2, D5⋊C8 [×2], C4.F5 [×2], C2×C5⋊C8 [×4], C2×C4×D5, C2×C4×D5 [×2], C2×D20, C2×D20 [×2], Q82D5 [×4], Q8×C10, D10⋊C8 [×4], C2×D5⋊C8, C2×C4.F5, C2×Q82D5, (C2×Q8).5F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C8○D4 [×2], C2×F5 [×3], (C22×C8)⋊C2, C22⋊F5 [×2], C22×F5, Q8.F5 [×2], C2×C22⋊F5, (C2×Q8).5F5

Smallest permutation representation of (C2×Q8).5F5
On 160 points
Generators in S160
(1 141)(2 142)(3 143)(4 144)(5 137)(6 138)(7 139)(8 140)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)(25 147)(26 148)(27 149)(28 150)(29 151)(30 152)(31 145)(32 146)(33 98)(34 99)(35 100)(36 101)(37 102)(38 103)(39 104)(40 97)(41 52)(42 53)(43 54)(44 55)(45 56)(46 49)(47 50)(48 51)(57 105)(58 106)(59 107)(60 108)(61 109)(62 110)(63 111)(64 112)(65 136)(66 129)(67 130)(68 131)(69 132)(70 133)(71 134)(72 135)(73 94)(74 95)(75 96)(76 89)(77 90)(78 91)(79 92)(80 93)(81 122)(82 123)(83 124)(84 125)(85 126)(86 127)(87 128)(88 121)(113 155)(114 156)(115 157)(116 158)(117 159)(118 160)(119 153)(120 154)
(1 22 5 18)(2 9 6 13)(3 24 7 20)(4 11 8 15)(10 143 14 139)(12 137 16 141)(17 144 21 140)(19 138 23 142)(25 47 29 43)(26 55 30 51)(27 41 31 45)(28 49 32 53)(33 136 37 132)(34 70 38 66)(35 130 39 134)(36 72 40 68)(42 150 46 146)(44 152 48 148)(50 151 54 147)(52 145 56 149)(57 155 61 159)(58 118 62 114)(59 157 63 153)(60 120 64 116)(65 102 69 98)(67 104 71 100)(73 123 77 127)(74 87 78 83)(75 125 79 121)(76 81 80 85)(82 90 86 94)(84 92 88 96)(89 122 93 126)(91 124 95 128)(97 131 101 135)(99 133 103 129)(105 113 109 117)(106 160 110 156)(107 115 111 119)(108 154 112 158)
(1 24 5 20)(2 17 6 21)(3 18 7 22)(4 19 8 23)(9 140 13 144)(10 141 14 137)(11 142 15 138)(12 143 16 139)(25 67 29 71)(26 68 30 72)(27 69 31 65)(28 70 32 66)(33 52 37 56)(34 53 38 49)(35 54 39 50)(36 55 40 51)(41 102 45 98)(42 103 46 99)(43 104 47 100)(44 97 48 101)(57 77 61 73)(58 78 62 74)(59 79 63 75)(60 80 64 76)(81 116 85 120)(82 117 86 113)(83 118 87 114)(84 119 88 115)(89 108 93 112)(90 109 94 105)(91 110 95 106)(92 111 96 107)(121 157 125 153)(122 158 126 154)(123 159 127 155)(124 160 128 156)(129 150 133 146)(130 151 134 147)(131 152 135 148)(132 145 136 149)
(1 92 132 50 113)(2 51 93 114 133)(3 115 52 134 94)(4 135 116 95 53)(5 96 136 54 117)(6 55 89 118 129)(7 119 56 130 90)(8 131 120 91 49)(9 26 126 58 103)(10 59 27 104 127)(11 97 60 128 28)(12 121 98 29 61)(13 30 122 62 99)(14 63 31 100 123)(15 101 64 124 32)(16 125 102 25 57)(17 36 112 83 146)(18 84 37 147 105)(19 148 85 106 38)(20 107 149 39 86)(21 40 108 87 150)(22 88 33 151 109)(23 152 81 110 34)(24 111 145 35 82)(41 71 73 143 157)(42 144 72 158 74)(43 159 137 75 65)(44 76 160 66 138)(45 67 77 139 153)(46 140 68 154 78)(47 155 141 79 69)(48 80 156 70 142)
(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)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H 5 8A···8H8I8J8K8L10A10B10C20A···20F
order122222224444444458···8888810101020···20
size11111010202022445555410···10202020204448···8

38 irreducible representations

dim1111111224448
type++++++++++
imageC1C2C2C2C2C4C4D4C8○D4F5C2×F5C22⋊F5Q8.F5
kernel(C2×Q8).5F5D10⋊C8C2×D5⋊C8C2×C4.F5C2×Q82D5C2×D20Q8×C10C4×D5C10C2×Q8C2×C4C4C2
# reps1411162481342

Matrix representation of (C2×Q8).5F5 in GL8(𝔽41)

10000000
01000000
004000000
000400000
00001000
00000100
00000010
00000001
,
040000000
10000000
004000000
00610000
00001000
00000100
00000010
00000001
,
09000000
90000000
004000000
000400000
000040000
000004000
000000400
000000040
,
10000000
01000000
00100000
00010000
000040404040
00001000
00000100
00000010
,
03000000
30000000
0035390000
003860000
00003501616
00001616035
00002519250
0000622226

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0],[0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,35,38,0,0,0,0,0,0,39,6,0,0,0,0,0,0,0,0,35,16,25,6,0,0,0,0,0,16,19,22,0,0,0,0,16,0,25,22,0,0,0,0,16,35,0,6] >;

(C2×Q8).5F5 in GAP, Magma, Sage, TeX

(C_2\times Q_8)._5F_5
% in TeX

G:=Group("(C2xQ8).5F5");
// GroupNames label

G:=SmallGroup(320,1125);
// by ID

G=gap.SmallGroup(320,1125);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,184,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^5=1,c^2=e^4=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,e*b*e^-1=a*b^-1,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations

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