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G = C42.68D6order 192 = 26·3

68th non-split extension by C42 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.68D6, C3⋊C83Q8, C34(C8⋊Q8), C4⋊C4.73D6, C4.32(S3×Q8), (C2×C12).84D4, C6.28(C4⋊Q8), C12.32(C2×Q8), C42.C2.3S3, C122Q8.16C2, C6.Q16.13C2, C2.20(D4⋊D6), C6.121(C8⋊C22), (C2×C12).382C23, (C4×C12).112C22, C42.S3.5C2, C12.Q8.14C2, C2.8(Dic3⋊Q8), C2.21(Q8.14D6), C6.122(C8.C22), C4⋊Dic3.152C22, (C2×C6).513(C2×D4), (C2×C4).65(C3⋊D4), (C2×C3⋊C8).125C22, (C3×C42.C2).2C2, (C3×C4⋊C4).120C22, (C2×C4).480(C22×S3), C22.186(C2×C3⋊D4), SmallGroup(192,623)

Series: Derived Chief Lower central Upper central

C1C2×C12 — C42.68D6
C1C3C6C2×C6C2×C12C2×C3⋊C8C42.S3 — C42.68D6
C3C6C2×C12 — C42.68D6
C1C22C42C42.C2

Generators and relations for C42.68D6
 G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=bc5 >

Subgroups: 224 in 90 conjugacy classes, 43 normal (27 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×6], C22, C6 [×3], C8 [×4], C2×C4 [×3], C2×C4 [×4], Q8 [×2], Dic3 [×2], C12 [×2], C12 [×4], C2×C6, C42, C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×2], C2×Q8, C3⋊C8 [×4], Dic6 [×2], C2×Dic3 [×2], C2×C12 [×3], C2×C12 [×2], C8⋊C4, C4.Q8 [×2], C2.D8 [×2], C42.C2, C4⋊Q8, C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C4⋊Dic3, C4×C12, C3×C4⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, C8⋊Q8, C42.S3, C6.Q16 [×2], C12.Q8 [×2], C122Q8, C3×C42.C2, C42.68D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×4], C23, D6 [×3], C2×D4, C2×Q8 [×2], C3⋊D4 [×2], C22×S3, C4⋊Q8, C8⋊C22, C8.C22, S3×Q8 [×2], C2×C3⋊D4, C8⋊Q8, Dic3⋊Q8, D4⋊D6, Q8.14D6, C42.68D6

Character table of C42.68D6

 class 12A2B2C34A4B4C4D4E4F4G4H6A6B6C8A8B8C8D12A12B12C12D12E12F12G12H12I12J
 size 111122244882424222121212124444448888
ρ1111111111111111111111111111111    trivial
ρ2111111111-1-1-1-11111111111111-1-1-1-1    linear of order 2
ρ31111111-1-11-11-11111-1-11-1-11-1-111-11-1    linear of order 2
ρ41111111-1-1-11-111111-1-11-1-11-1-11-11-11    linear of order 2
ρ51111111-1-1-111-1111-111-1-1-11-1-11-11-11    linear of order 2
ρ61111111-1-11-1-11111-111-1-1-11-1-111-11-1    linear of order 2
ρ711111111111-1-1111-1-1-1-11111111111    linear of order 2
ρ8111111111-1-111111-1-1-1-1111111-1-1-1-1    linear of order 2
ρ922222-2-22-20000222000022-2-2-2-20000    orthogonal lifted from D4
ρ102222-122222200-1-1-10000-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ112222-122-2-2-2200-1-1-1000011-111-11-11-1    orthogonal lifted from D6
ρ122222-122-2-22-200-1-1-1000011-111-1-11-11    orthogonal lifted from D6
ρ132222-12222-2-200-1-1-10000-1-1-1-1-1-11111    orthogonal lifted from D6
ρ1422222-2-2-2200002220000-2-2-222-20000    orthogonal lifted from D4
ρ1522-2-22-22000000-2-220-22000200-20000    symplectic lifted from Q8, Schur index 2
ρ1622-2-222-2000000-2-22-200200-20020000    symplectic lifted from Q8, Schur index 2
ρ1722-2-22-22000000-2-2202-2000200-20000    symplectic lifted from Q8, Schur index 2
ρ1822-2-222-2000000-2-22200-200-20020000    symplectic lifted from Q8, Schur index 2
ρ192222-1-2-2-220000-1-1-10000111-1-11-3--3--3-3    complex lifted from C3⋊D4
ρ202222-1-2-22-20000-1-1-10000-1-11111-3-3--3--3    complex lifted from C3⋊D4
ρ212222-1-2-22-20000-1-1-10000-1-11111--3--3-3-3    complex lifted from C3⋊D4
ρ222222-1-2-2-220000-1-1-10000111-1-11--3-3-3--3    complex lifted from C3⋊D4
ρ234-44-4-2000000002-22000000023-2300000    orthogonal lifted from D4⋊D6
ρ244-44-4400000000-44-400000000000000    orthogonal lifted from C8⋊C22
ρ254-44-4-2000000002-220000000-232300000    orthogonal lifted from D4⋊D6
ρ2644-4-4-24-400000022-2000000200-20000    symplectic lifted from S3×Q8, Schur index 2
ρ274-4-44-200000000-2220000-232300000000    symplectic lifted from Q8.14D6, Schur index 2
ρ284-4-444000000004-4-400000000000000    symplectic lifted from C8.C22, Schur index 2
ρ294-4-44-200000000-222000023-2300000000    symplectic lifted from Q8.14D6, Schur index 2
ρ3044-4-4-2-4400000022-2000000-20020000    symplectic lifted from S3×Q8, Schur index 2

Smallest permutation representation of C42.68D6
Regular action on 192 points
Generators in S192
(1 158 7 164)(2 104 8 98)(3 160 9 166)(4 106 10 100)(5 162 11 168)(6 108 12 102)(13 90 19 96)(14 76 20 82)(15 92 21 86)(16 78 22 84)(17 94 23 88)(18 80 24 74)(25 48 31 42)(26 62 32 68)(27 38 33 44)(28 64 34 70)(29 40 35 46)(30 66 36 72)(37 126 43 132)(39 128 45 122)(41 130 47 124)(49 148 55 154)(50 182 56 188)(51 150 57 156)(52 184 58 190)(53 152 59 146)(54 186 60 192)(61 125 67 131)(63 127 69 121)(65 129 71 123)(73 174 79 180)(75 176 81 170)(77 178 83 172)(85 171 91 177)(87 173 93 179)(89 175 95 169)(97 118 103 112)(99 120 105 114)(101 110 107 116)(109 161 115 167)(111 163 117 157)(113 165 119 159)(133 145 139 151)(134 191 140 185)(135 147 141 153)(136 181 142 187)(137 149 143 155)(138 183 144 189)
(1 51 112 144)(2 133 113 52)(3 53 114 134)(4 135 115 54)(5 55 116 136)(6 137 117 56)(7 57 118 138)(8 139 119 58)(9 59 120 140)(10 141 109 60)(11 49 110 142)(12 143 111 50)(13 48 176 67)(14 68 177 37)(15 38 178 69)(16 70 179 39)(17 40 180 71)(18 72 169 41)(19 42 170 61)(20 62 171 43)(21 44 172 63)(22 64 173 45)(23 46 174 65)(24 66 175 47)(25 75 125 96)(26 85 126 76)(27 77 127 86)(28 87 128 78)(29 79 129 88)(30 89 130 80)(31 81 131 90)(32 91 132 82)(33 83 121 92)(34 93 122 84)(35 73 123 94)(36 95 124 74)(97 189 158 150)(98 151 159 190)(99 191 160 152)(100 153 161 192)(101 181 162 154)(102 155 163 182)(103 183 164 156)(104 145 165 184)(105 185 166 146)(106 147 167 186)(107 187 168 148)(108 149 157 188)
(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)(157 158 159 160 161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192)
(1 180 7 174)(2 64 8 70)(3 178 9 172)(4 62 10 68)(5 176 11 170)(6 72 12 66)(13 110 19 116)(14 135 20 141)(15 120 21 114)(16 133 22 139)(17 118 23 112)(18 143 24 137)(25 148 31 154)(26 100 32 106)(27 146 33 152)(28 98 34 104)(29 156 35 150)(30 108 36 102)(37 115 43 109)(38 59 44 53)(39 113 45 119)(40 57 46 51)(41 111 47 117)(42 55 48 49)(50 175 56 169)(52 173 58 179)(54 171 60 177)(61 136 67 142)(63 134 69 140)(65 144 71 138)(73 158 79 164)(74 155 80 149)(75 168 81 162)(76 153 82 147)(77 166 83 160)(78 151 84 145)(85 192 91 186)(86 105 92 99)(87 190 93 184)(88 103 94 97)(89 188 95 182)(90 101 96 107)(121 191 127 185)(122 165 128 159)(123 189 129 183)(124 163 130 157)(125 187 131 181)(126 161 132 167)

G:=sub<Sym(192)| (1,158,7,164)(2,104,8,98)(3,160,9,166)(4,106,10,100)(5,162,11,168)(6,108,12,102)(13,90,19,96)(14,76,20,82)(15,92,21,86)(16,78,22,84)(17,94,23,88)(18,80,24,74)(25,48,31,42)(26,62,32,68)(27,38,33,44)(28,64,34,70)(29,40,35,46)(30,66,36,72)(37,126,43,132)(39,128,45,122)(41,130,47,124)(49,148,55,154)(50,182,56,188)(51,150,57,156)(52,184,58,190)(53,152,59,146)(54,186,60,192)(61,125,67,131)(63,127,69,121)(65,129,71,123)(73,174,79,180)(75,176,81,170)(77,178,83,172)(85,171,91,177)(87,173,93,179)(89,175,95,169)(97,118,103,112)(99,120,105,114)(101,110,107,116)(109,161,115,167)(111,163,117,157)(113,165,119,159)(133,145,139,151)(134,191,140,185)(135,147,141,153)(136,181,142,187)(137,149,143,155)(138,183,144,189), (1,51,112,144)(2,133,113,52)(3,53,114,134)(4,135,115,54)(5,55,116,136)(6,137,117,56)(7,57,118,138)(8,139,119,58)(9,59,120,140)(10,141,109,60)(11,49,110,142)(12,143,111,50)(13,48,176,67)(14,68,177,37)(15,38,178,69)(16,70,179,39)(17,40,180,71)(18,72,169,41)(19,42,170,61)(20,62,171,43)(21,44,172,63)(22,64,173,45)(23,46,174,65)(24,66,175,47)(25,75,125,96)(26,85,126,76)(27,77,127,86)(28,87,128,78)(29,79,129,88)(30,89,130,80)(31,81,131,90)(32,91,132,82)(33,83,121,92)(34,93,122,84)(35,73,123,94)(36,95,124,74)(97,189,158,150)(98,151,159,190)(99,191,160,152)(100,153,161,192)(101,181,162,154)(102,155,163,182)(103,183,164,156)(104,145,165,184)(105,185,166,146)(106,147,167,186)(107,187,168,148)(108,149,157,188), (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)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,180,7,174)(2,64,8,70)(3,178,9,172)(4,62,10,68)(5,176,11,170)(6,72,12,66)(13,110,19,116)(14,135,20,141)(15,120,21,114)(16,133,22,139)(17,118,23,112)(18,143,24,137)(25,148,31,154)(26,100,32,106)(27,146,33,152)(28,98,34,104)(29,156,35,150)(30,108,36,102)(37,115,43,109)(38,59,44,53)(39,113,45,119)(40,57,46,51)(41,111,47,117)(42,55,48,49)(50,175,56,169)(52,173,58,179)(54,171,60,177)(61,136,67,142)(63,134,69,140)(65,144,71,138)(73,158,79,164)(74,155,80,149)(75,168,81,162)(76,153,82,147)(77,166,83,160)(78,151,84,145)(85,192,91,186)(86,105,92,99)(87,190,93,184)(88,103,94,97)(89,188,95,182)(90,101,96,107)(121,191,127,185)(122,165,128,159)(123,189,129,183)(124,163,130,157)(125,187,131,181)(126,161,132,167)>;

G:=Group( (1,158,7,164)(2,104,8,98)(3,160,9,166)(4,106,10,100)(5,162,11,168)(6,108,12,102)(13,90,19,96)(14,76,20,82)(15,92,21,86)(16,78,22,84)(17,94,23,88)(18,80,24,74)(25,48,31,42)(26,62,32,68)(27,38,33,44)(28,64,34,70)(29,40,35,46)(30,66,36,72)(37,126,43,132)(39,128,45,122)(41,130,47,124)(49,148,55,154)(50,182,56,188)(51,150,57,156)(52,184,58,190)(53,152,59,146)(54,186,60,192)(61,125,67,131)(63,127,69,121)(65,129,71,123)(73,174,79,180)(75,176,81,170)(77,178,83,172)(85,171,91,177)(87,173,93,179)(89,175,95,169)(97,118,103,112)(99,120,105,114)(101,110,107,116)(109,161,115,167)(111,163,117,157)(113,165,119,159)(133,145,139,151)(134,191,140,185)(135,147,141,153)(136,181,142,187)(137,149,143,155)(138,183,144,189), (1,51,112,144)(2,133,113,52)(3,53,114,134)(4,135,115,54)(5,55,116,136)(6,137,117,56)(7,57,118,138)(8,139,119,58)(9,59,120,140)(10,141,109,60)(11,49,110,142)(12,143,111,50)(13,48,176,67)(14,68,177,37)(15,38,178,69)(16,70,179,39)(17,40,180,71)(18,72,169,41)(19,42,170,61)(20,62,171,43)(21,44,172,63)(22,64,173,45)(23,46,174,65)(24,66,175,47)(25,75,125,96)(26,85,126,76)(27,77,127,86)(28,87,128,78)(29,79,129,88)(30,89,130,80)(31,81,131,90)(32,91,132,82)(33,83,121,92)(34,93,122,84)(35,73,123,94)(36,95,124,74)(97,189,158,150)(98,151,159,190)(99,191,160,152)(100,153,161,192)(101,181,162,154)(102,155,163,182)(103,183,164,156)(104,145,165,184)(105,185,166,146)(106,147,167,186)(107,187,168,148)(108,149,157,188), (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)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,180,7,174)(2,64,8,70)(3,178,9,172)(4,62,10,68)(5,176,11,170)(6,72,12,66)(13,110,19,116)(14,135,20,141)(15,120,21,114)(16,133,22,139)(17,118,23,112)(18,143,24,137)(25,148,31,154)(26,100,32,106)(27,146,33,152)(28,98,34,104)(29,156,35,150)(30,108,36,102)(37,115,43,109)(38,59,44,53)(39,113,45,119)(40,57,46,51)(41,111,47,117)(42,55,48,49)(50,175,56,169)(52,173,58,179)(54,171,60,177)(61,136,67,142)(63,134,69,140)(65,144,71,138)(73,158,79,164)(74,155,80,149)(75,168,81,162)(76,153,82,147)(77,166,83,160)(78,151,84,145)(85,192,91,186)(86,105,92,99)(87,190,93,184)(88,103,94,97)(89,188,95,182)(90,101,96,107)(121,191,127,185)(122,165,128,159)(123,189,129,183)(124,163,130,157)(125,187,131,181)(126,161,132,167) );

G=PermutationGroup([(1,158,7,164),(2,104,8,98),(3,160,9,166),(4,106,10,100),(5,162,11,168),(6,108,12,102),(13,90,19,96),(14,76,20,82),(15,92,21,86),(16,78,22,84),(17,94,23,88),(18,80,24,74),(25,48,31,42),(26,62,32,68),(27,38,33,44),(28,64,34,70),(29,40,35,46),(30,66,36,72),(37,126,43,132),(39,128,45,122),(41,130,47,124),(49,148,55,154),(50,182,56,188),(51,150,57,156),(52,184,58,190),(53,152,59,146),(54,186,60,192),(61,125,67,131),(63,127,69,121),(65,129,71,123),(73,174,79,180),(75,176,81,170),(77,178,83,172),(85,171,91,177),(87,173,93,179),(89,175,95,169),(97,118,103,112),(99,120,105,114),(101,110,107,116),(109,161,115,167),(111,163,117,157),(113,165,119,159),(133,145,139,151),(134,191,140,185),(135,147,141,153),(136,181,142,187),(137,149,143,155),(138,183,144,189)], [(1,51,112,144),(2,133,113,52),(3,53,114,134),(4,135,115,54),(5,55,116,136),(6,137,117,56),(7,57,118,138),(8,139,119,58),(9,59,120,140),(10,141,109,60),(11,49,110,142),(12,143,111,50),(13,48,176,67),(14,68,177,37),(15,38,178,69),(16,70,179,39),(17,40,180,71),(18,72,169,41),(19,42,170,61),(20,62,171,43),(21,44,172,63),(22,64,173,45),(23,46,174,65),(24,66,175,47),(25,75,125,96),(26,85,126,76),(27,77,127,86),(28,87,128,78),(29,79,129,88),(30,89,130,80),(31,81,131,90),(32,91,132,82),(33,83,121,92),(34,93,122,84),(35,73,123,94),(36,95,124,74),(97,189,158,150),(98,151,159,190),(99,191,160,152),(100,153,161,192),(101,181,162,154),(102,155,163,182),(103,183,164,156),(104,145,165,184),(105,185,166,146),(106,147,167,186),(107,187,168,148),(108,149,157,188)], [(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),(157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192)], [(1,180,7,174),(2,64,8,70),(3,178,9,172),(4,62,10,68),(5,176,11,170),(6,72,12,66),(13,110,19,116),(14,135,20,141),(15,120,21,114),(16,133,22,139),(17,118,23,112),(18,143,24,137),(25,148,31,154),(26,100,32,106),(27,146,33,152),(28,98,34,104),(29,156,35,150),(30,108,36,102),(37,115,43,109),(38,59,44,53),(39,113,45,119),(40,57,46,51),(41,111,47,117),(42,55,48,49),(50,175,56,169),(52,173,58,179),(54,171,60,177),(61,136,67,142),(63,134,69,140),(65,144,71,138),(73,158,79,164),(74,155,80,149),(75,168,81,162),(76,153,82,147),(77,166,83,160),(78,151,84,145),(85,192,91,186),(86,105,92,99),(87,190,93,184),(88,103,94,97),(89,188,95,182),(90,101,96,107),(121,191,127,185),(122,165,128,159),(123,189,129,183),(124,163,130,157),(125,187,131,181),(126,161,132,167)])

Matrix representation of C42.68D6 in GL6(𝔽73)

72160000
910000
0071400
00596600
0000714
00005966
,
100000
010000
00720710
00072071
001010
000101
,
120000
72720000
008303321
0043515212
0045176543
0056283022
,
72710000
110000
00134658
0033725267
002687239
006671401

G:=sub<GL(6,GF(73))| [72,9,0,0,0,0,16,1,0,0,0,0,0,0,7,59,0,0,0,0,14,66,0,0,0,0,0,0,7,59,0,0,0,0,14,66],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,1,0,0,0,0,72,0,1,0,0,71,0,1,0,0,0,0,71,0,1],[1,72,0,0,0,0,2,72,0,0,0,0,0,0,8,43,45,56,0,0,30,51,17,28,0,0,33,52,65,30,0,0,21,12,43,22],[72,1,0,0,0,0,71,1,0,0,0,0,0,0,1,33,2,66,0,0,34,72,68,71,0,0,6,52,72,40,0,0,58,67,39,1] >;

C42.68D6 in GAP, Magma, Sage, TeX

C_4^2._{68}D_6
% in TeX

G:=Group("C4^2.68D6");
// GroupNames label

G:=SmallGroup(192,623);
// by ID

G=gap.SmallGroup(192,623);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,477,64,422,471,58,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b*c^5>;
// generators/relations

Export

Character table of C42.68D6 in TeX

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