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## G = C10.1182+ 1+4order 320 = 26·5

### 27th 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 — C10.1182+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C5⋊D4 — C4×C5⋊D4 — C10.1182+ 1+4
 Lower central C5 — C2×C10 — C10.1182+ 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C10.1182+ 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, e2=a5, ab=ba, cac=a-1, ad=da, ae=ea, cbc=a5b-1, bd=db, ebe-1=a5b, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 742 in 228 conjugacy classes, 107 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C4×D4, C4×Q8, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×Q8, C22×D5, C22×C10, D43Q8, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C22×C20, Q8×C10, Dic5.14D4, Dic54D4, C20⋊Q8, Dic5.Q8, D5×C4⋊C4, D102Q8, C2×C4⋊Dic5, C4×C5⋊D4, Q8×Dic5, D103Q8, C5×C22⋊Q8, C10.1182+ 1+4
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C4○D4, C24, D10, C22×Q8, C2×C4○D4, 2+ 1+4, C22×D5, D43Q8, D42D5, Q8×D5, C23×D5, C2×D42D5, C2×Q8×D5, D48D10, C10.1182+ 1+4

Smallest permutation representation of C10.1182+ 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)
(1 145 24 133)(2 146 25 134)(3 147 26 135)(4 148 27 136)(5 149 28 137)(6 150 29 138)(7 141 30 139)(8 142 21 140)(9 143 22 131)(10 144 23 132)(11 33 153 45)(12 34 154 46)(13 35 155 47)(14 36 156 48)(15 37 157 49)(16 38 158 50)(17 39 159 41)(18 40 160 42)(19 31 151 43)(20 32 152 44)(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)
(2 10)(3 9)(4 8)(5 7)(11 158)(12 157)(13 156)(14 155)(15 154)(16 153)(17 152)(18 151)(19 160)(20 159)(21 27)(22 26)(23 25)(28 30)(31 35)(32 34)(36 40)(37 39)(41 49)(42 48)(43 47)(44 46)(51 55)(52 54)(56 60)(57 59)(61 69)(62 68)(63 67)(64 66)(71 75)(72 74)(76 80)(77 79)(81 89)(82 88)(83 87)(84 86)(91 102)(92 101)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(111 122)(112 121)(113 130)(114 129)(115 128)(116 127)(117 126)(118 125)(119 124)(120 123)(131 142)(132 141)(133 150)(134 149)(135 148)(136 147)(137 146)(138 145)(139 144)(140 143)
(1 65 24 53)(2 66 25 54)(3 67 26 55)(4 68 27 56)(5 69 28 57)(6 70 29 58)(7 61 30 59)(8 62 21 60)(9 63 22 51)(10 64 23 52)(11 113 153 125)(12 114 154 126)(13 115 155 127)(14 116 156 128)(15 117 157 129)(16 118 158 130)(17 119 159 121)(18 120 160 122)(19 111 151 123)(20 112 152 124)(31 71 43 83)(32 72 44 84)(33 73 45 85)(34 74 46 86)(35 75 47 87)(36 76 48 88)(37 77 49 89)(38 78 50 90)(39 79 41 81)(40 80 42 82)(91 131 103 143)(92 132 104 144)(93 133 105 145)(94 134 106 146)(95 135 107 147)(96 136 108 148)(97 137 109 149)(98 138 110 150)(99 139 101 141)(100 140 102 142)
(1 78 6 73)(2 79 7 74)(3 80 8 75)(4 71 9 76)(5 72 10 77)(11 110 16 105)(12 101 17 106)(13 102 18 107)(14 103 19 108)(15 104 20 109)(21 87 26 82)(22 88 27 83)(23 89 28 84)(24 90 29 85)(25 81 30 86)(31 63 36 68)(32 64 37 69)(33 65 38 70)(34 66 39 61)(35 67 40 62)(41 59 46 54)(42 60 47 55)(43 51 48 56)(44 52 49 57)(45 53 50 58)(91 151 96 156)(92 152 97 157)(93 153 98 158)(94 154 99 159)(95 155 100 160)(111 136 116 131)(112 137 117 132)(113 138 118 133)(114 139 119 134)(115 140 120 135)(121 146 126 141)(122 147 127 142)(123 148 128 143)(124 149 129 144)(125 150 130 145)```

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

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

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

53 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C ··· 4G 4H ··· 4M 4N 4O 4P 4Q 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I ··· 20P order 1 2 2 2 2 2 2 2 4 4 4 ··· 4 4 ··· 4 4 4 4 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 10 10 2 2 4 ··· 4 10 ··· 10 20 20 20 20 2 2 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + - + + + + + + - - + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 Q8 D5 C4○D4 D10 D10 D10 D10 2+ 1+4 D4⋊2D5 Q8×D5 D4⋊8D10 kernel C10.1182+ 1+4 Dic5.14D4 Dic5⋊4D4 C20⋊Q8 Dic5.Q8 D5×C4⋊C4 D10⋊2Q8 C2×C4⋊Dic5 C4×C5⋊D4 Q8×Dic5 D10⋊3Q8 C5×C22⋊Q8 C5⋊D4 C22⋊Q8 C20 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C10 C4 C22 C2 # reps 1 2 2 1 2 1 2 1 1 1 1 1 4 2 4 4 6 2 2 1 4 4 4

Matrix representation of C10.1182+ 1+4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 7 0 0 0 0 34 7 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 30 40 0 0 0 0 40 11
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 34 7 0 0 0 0 40 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 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 30 40 0 0 0 0 40 11
,
 32 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 11 0 0 0 0 11 1

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,34,0,0,0,0,7,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,40,0,0,0,0,40,11],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,34,40,0,0,0,0,7,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,40,0,0,0,0,40,11],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,11,0,0,0,0,11,1] >;`

C10.1182+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{118}2_+^{1+4}`
`% in TeX`

`G:=Group("C10.118ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1307);`
`// by ID`

`G=gap.SmallGroup(320,1307);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,100,570,185,192,12550]);`
`// Polycyclic`

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