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## G = C10.792- 1+4order 320 = 26·5

### 34th non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C4○D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C10.792- 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×D4⋊2D5 — C10.792- 1+4
 Lower central C5 — C2×C10 — C10.792- 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C10.792- 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=e2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, ebe-1=a5b, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 926 in 270 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C42, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C42⋊C2, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C4⋊Q8, C22×Q8, C2×C4○D4, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C23.38C23, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C23.D5, C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C2×Dic10, C2×C4×D5, D42D5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, C23.11D10, Dic5.14D4, Dic5.5D4, C22.D20, C20⋊Q8, D10⋊Q8, C23.23D10, C23.18D10, C5×C22.D4, C22×Dic10, C2×D42D5, C10.792- 1+4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2- 1+4, C22×D5, C23.38C23, D4×D5, C23×D5, C2×D4×D5, D4.10D10, C10.792- 1+4

Smallest permutation representation of C10.792- 1+4
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)
(1 150 30 138)(2 141 21 139)(3 142 22 140)(4 143 23 131)(5 144 24 132)(6 145 25 133)(7 146 26 134)(8 147 27 135)(9 148 28 136)(10 149 29 137)(11 45 153 33)(12 46 154 34)(13 47 155 35)(14 48 156 36)(15 49 157 37)(16 50 158 38)(17 41 159 39)(18 42 160 40)(19 43 151 31)(20 44 152 32)(51 103 63 91)(52 104 64 92)(53 105 65 93)(54 106 66 94)(55 107 67 95)(56 108 68 96)(57 109 69 97)(58 110 70 98)(59 101 61 99)(60 102 62 100)(71 123 83 111)(72 124 84 112)(73 125 85 113)(74 126 86 114)(75 127 87 115)(76 128 88 116)(77 129 89 117)(78 130 90 118)(79 121 81 119)(80 122 82 120)
(1 98)(2 99)(3 100)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 81)(18 82)(19 83)(20 84)(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)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)
(1 138 30 150)(2 137 21 149)(3 136 22 148)(4 135 23 147)(5 134 24 146)(6 133 25 145)(7 132 26 144)(8 131 27 143)(9 140 28 142)(10 139 29 141)(11 50 153 38)(12 49 154 37)(13 48 155 36)(14 47 156 35)(15 46 157 34)(16 45 158 33)(17 44 159 32)(18 43 160 31)(19 42 151 40)(20 41 152 39)(51 107 63 95)(52 106 64 94)(53 105 65 93)(54 104 66 92)(55 103 67 91)(56 102 68 100)(57 101 69 99)(58 110 70 98)(59 109 61 97)(60 108 62 96)(71 120 83 122)(72 119 84 121)(73 118 85 130)(74 117 86 129)(75 116 87 128)(76 115 88 127)(77 114 89 126)(78 113 90 125)(79 112 81 124)(80 111 82 123)
(1 50 30 38)(2 41 21 39)(3 42 22 40)(4 43 23 31)(5 44 24 32)(6 45 25 33)(7 46 26 34)(8 47 27 35)(9 48 28 36)(10 49 29 37)(11 150 153 138)(12 141 154 139)(13 142 155 140)(14 143 156 131)(15 144 157 132)(16 145 158 133)(17 146 159 134)(18 147 160 135)(19 148 151 136)(20 149 152 137)(51 88 63 76)(52 89 64 77)(53 90 65 78)(54 81 66 79)(55 82 67 80)(56 83 68 71)(57 84 69 72)(58 85 70 73)(59 86 61 74)(60 87 62 75)(91 123 103 111)(92 124 104 112)(93 125 105 113)(94 126 106 114)(95 127 107 115)(96 128 108 116)(97 129 109 117)(98 130 110 118)(99 121 101 119)(100 122 102 120)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4E 4F 4G 4H 4I 4J ··· 4N 5A 5B 10A ··· 10F 10G 10H 10I 10J 10K 10L 20A ··· 20H 20I ··· 20N order 1 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 10 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 4 20 4 ··· 4 10 10 10 10 20 ··· 20 2 2 2 ··· 2 4 4 4 4 8 8 4 ··· 4 8 ··· 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 D10 D10 D10 D10 2- 1+4 D4×D5 D4.10D10 kernel C10.792- 1+4 C23.11D10 Dic5.14D4 Dic5.5D4 C22.D20 C20⋊Q8 D10⋊Q8 C23.23D10 C23.18D10 C5×C22.D4 C22×Dic10 C2×D4⋊2D5 C2×Dic5 C22.D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 C22 C2 # reps 1 1 2 2 1 2 2 1 1 1 1 1 4 2 6 4 2 2 2 4 8

Matrix representation of C10.792- 1+4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 35 34 0 0 0 0 6 0 0 0 0 0 0 0 34 34 0 0 0 0 7 1
,
 40 0 0 0 0 0 15 1 0 0 0 0 0 0 17 0 0 39 0 0 0 17 2 2 0 0 19 19 24 0 0 0 22 0 0 24
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 39 9 35 20 0 0 4 2 26 15 0 0 6 33 30 32 0 0 6 14 9 11
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 21 4 27 39 0 0 3 20 12 2 0 0 22 22 17 0 0 0 32 10 4 24
,
 24 36 0 0 0 0 33 17 0 0 0 0 0 0 1 0 0 28 0 0 0 1 13 13 0 0 3 3 40 0 0 0 38 0 0 40

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,34,0,0,0,0,0,0,0,34,7,0,0,0,0,34,1],[40,15,0,0,0,0,0,1,0,0,0,0,0,0,17,0,19,22,0,0,0,17,19,0,0,0,0,2,24,0,0,0,39,2,0,24],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,39,4,6,6,0,0,9,2,33,14,0,0,35,26,30,9,0,0,20,15,32,11],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,21,3,22,32,0,0,4,20,22,10,0,0,27,12,17,4,0,0,39,2,0,24],[24,33,0,0,0,0,36,17,0,0,0,0,0,0,1,0,3,38,0,0,0,1,3,0,0,0,0,13,40,0,0,0,28,13,0,40] >;`

C10.792- 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{79}2_-^{1+4}`
`% in TeX`

`G:=Group("C10.79ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1320);`
`// by ID`

`G=gap.SmallGroup(320,1320);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,387,1123,185,136,12550]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^10=b^4=c^2=1,d^2=e^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=a^5*b,c*d=d*c,c*e=e*c,e*d*e^-1=b^2*d>;`
`// generators/relations`

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