metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊8Q8, C42.169D6, C6.332- (1+4), C4⋊Q8.14S3, C4.17(S3×Q8), C4⋊C4.121D6, C3⋊5(Q8⋊3Q8), C12.52(C2×Q8), (C2×Q8).165D6, C6.44(C22×Q8), (C2×C12).98C23, (C2×C6).265C24, Dic3.14(C2×Q8), (C4×Dic6).25C2, (Q8×Dic3).13C2, Dic3.Q8.4C2, C12.135(C4○D4), C4.18(D4⋊2S3), (C4×C12).206C22, Dic3⋊Q8.8C2, C12.3Q8.14C2, (C6×Q8).132C22, Dic6⋊C4.12C2, Dic3⋊C4.57C22, C4⋊Dic3.382C22, C22.286(S3×C23), C2.34(Q8.15D6), (C4×Dic3).157C22, (C2×Dic3).270C23, (C2×Dic6).302C22, C2.27(C2×S3×Q8), C6.99(C2×C4○D4), (C3×C4⋊Q8).14C2, C2.63(C2×D4⋊2S3), (C2×C4).90(C22×S3), (C3×C4⋊C4).208C22, SmallGroup(192,1280)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 384 in 200 conjugacy classes, 107 normal (27 characteristic)
C1, C2 [×3], C3, C4 [×4], C4 [×15], C22, C6 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], Q8 [×10], Dic3 [×4], Dic3 [×6], C12 [×4], C12 [×5], C2×C6, C42, C42 [×8], C4⋊C4 [×4], C4⋊C4 [×18], C2×Q8 [×2], C2×Q8 [×2], Dic6 [×4], Dic6 [×2], C2×Dic3 [×8], C2×C12 [×3], C2×C12 [×4], C3×Q8 [×4], C4×Q8 [×6], C42.C2 [×6], C4⋊Q8, C4⋊Q8 [×2], C4×Dic3 [×8], Dic3⋊C4 [×12], C4⋊Dic3 [×2], C4⋊Dic3 [×4], C4×C12, C3×C4⋊C4 [×4], C2×Dic6 [×2], C6×Q8 [×2], Q8⋊3Q8, C4×Dic6 [×2], Dic6⋊C4 [×2], Dic3.Q8 [×4], C12.3Q8 [×2], Dic3⋊Q8 [×2], Q8×Dic3 [×2], C3×C4⋊Q8, Dic6⋊8Q8
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×Q8, C2×C4○D4, 2- (1+4), D4⋊2S3 [×2], S3×Q8 [×2], S3×C23, Q8⋊3Q8, C2×D4⋊2S3, C2×S3×Q8, Q8.15D6, Dic6⋊8Q8
Generators and relations
G = < a,b,c,d | a12=c4=1, b2=a6, d2=c2, bab-1=a-1, ac=ca, dad-1=a7, bc=cb, bd=db, dcd-1=c-1 >
►Regular action on 192 points
Generators in S192
(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 82 7 76)(2 81 8 75)(3 80 9 74)(4 79 10 73)(5 78 11 84)(6 77 12 83)(13 47 19 41)(14 46 20 40)(15 45 21 39)(16 44 22 38)(17 43 23 37)(18 42 24 48)(25 181 31 187)(26 192 32 186)(27 191 33 185)(28 190 34 184)(29 189 35 183)(30 188 36 182)(49 62 55 68)(50 61 56 67)(51 72 57 66)(52 71 58 65)(53 70 59 64)(54 69 60 63)(85 152 91 146)(86 151 92 145)(87 150 93 156)(88 149 94 155)(89 148 95 154)(90 147 96 153)(97 172 103 178)(98 171 104 177)(99 170 105 176)(100 169 106 175)(101 180 107 174)(102 179 108 173)(109 167 115 161)(110 166 116 160)(111 165 117 159)(112 164 118 158)(113 163 119 157)(114 162 120 168)(121 136 127 142)(122 135 128 141)(123 134 129 140)(124 133 130 139)(125 144 131 138)(126 143 132 137)
(1 148 112 99)(2 149 113 100)(3 150 114 101)(4 151 115 102)(5 152 116 103)(6 153 117 104)(7 154 118 105)(8 155 119 106)(9 156 120 107)(10 145 109 108)(11 146 110 97)(12 147 111 98)(13 125 192 54)(14 126 181 55)(15 127 182 56)(16 128 183 57)(17 129 184 58)(18 130 185 59)(19 131 186 60)(20 132 187 49)(21 121 188 50)(22 122 189 51)(23 123 190 52)(24 124 191 53)(25 62 40 137)(26 63 41 138)(27 64 42 139)(28 65 43 140)(29 66 44 141)(30 67 45 142)(31 68 46 143)(32 69 47 144)(33 70 48 133)(34 71 37 134)(35 72 38 135)(36 61 39 136)(73 86 167 173)(74 87 168 174)(75 88 157 175)(76 89 158 176)(77 90 159 177)(78 91 160 178)(79 92 161 179)(80 93 162 180)(81 94 163 169)(82 95 164 170)(83 96 165 171)(84 85 166 172)
(1 16 112 183)(2 23 113 190)(3 18 114 185)(4 13 115 192)(5 20 116 187)(6 15 117 182)(7 22 118 189)(8 17 119 184)(9 24 120 191)(10 19 109 186)(11 14 110 181)(12 21 111 188)(25 78 40 160)(26 73 41 167)(27 80 42 162)(28 75 43 157)(29 82 44 164)(30 77 45 159)(31 84 46 166)(32 79 47 161)(33 74 48 168)(34 81 37 163)(35 76 38 158)(36 83 39 165)(49 103 132 152)(50 98 121 147)(51 105 122 154)(52 100 123 149)(53 107 124 156)(54 102 125 151)(55 97 126 146)(56 104 127 153)(57 99 128 148)(58 106 129 155)(59 101 130 150)(60 108 131 145)(61 171 136 96)(62 178 137 91)(63 173 138 86)(64 180 139 93)(65 175 140 88)(66 170 141 95)(67 177 142 90)(68 172 143 85)(69 179 144 92)(70 174 133 87)(71 169 134 94)(72 176 135 89)
G:=sub<Sym(192)| (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), (1,82,7,76)(2,81,8,75)(3,80,9,74)(4,79,10,73)(5,78,11,84)(6,77,12,83)(13,47,19,41)(14,46,20,40)(15,45,21,39)(16,44,22,38)(17,43,23,37)(18,42,24,48)(25,181,31,187)(26,192,32,186)(27,191,33,185)(28,190,34,184)(29,189,35,183)(30,188,36,182)(49,62,55,68)(50,61,56,67)(51,72,57,66)(52,71,58,65)(53,70,59,64)(54,69,60,63)(85,152,91,146)(86,151,92,145)(87,150,93,156)(88,149,94,155)(89,148,95,154)(90,147,96,153)(97,172,103,178)(98,171,104,177)(99,170,105,176)(100,169,106,175)(101,180,107,174)(102,179,108,173)(109,167,115,161)(110,166,116,160)(111,165,117,159)(112,164,118,158)(113,163,119,157)(114,162,120,168)(121,136,127,142)(122,135,128,141)(123,134,129,140)(124,133,130,139)(125,144,131,138)(126,143,132,137), (1,148,112,99)(2,149,113,100)(3,150,114,101)(4,151,115,102)(5,152,116,103)(6,153,117,104)(7,154,118,105)(8,155,119,106)(9,156,120,107)(10,145,109,108)(11,146,110,97)(12,147,111,98)(13,125,192,54)(14,126,181,55)(15,127,182,56)(16,128,183,57)(17,129,184,58)(18,130,185,59)(19,131,186,60)(20,132,187,49)(21,121,188,50)(22,122,189,51)(23,123,190,52)(24,124,191,53)(25,62,40,137)(26,63,41,138)(27,64,42,139)(28,65,43,140)(29,66,44,141)(30,67,45,142)(31,68,46,143)(32,69,47,144)(33,70,48,133)(34,71,37,134)(35,72,38,135)(36,61,39,136)(73,86,167,173)(74,87,168,174)(75,88,157,175)(76,89,158,176)(77,90,159,177)(78,91,160,178)(79,92,161,179)(80,93,162,180)(81,94,163,169)(82,95,164,170)(83,96,165,171)(84,85,166,172), (1,16,112,183)(2,23,113,190)(3,18,114,185)(4,13,115,192)(5,20,116,187)(6,15,117,182)(7,22,118,189)(8,17,119,184)(9,24,120,191)(10,19,109,186)(11,14,110,181)(12,21,111,188)(25,78,40,160)(26,73,41,167)(27,80,42,162)(28,75,43,157)(29,82,44,164)(30,77,45,159)(31,84,46,166)(32,79,47,161)(33,74,48,168)(34,81,37,163)(35,76,38,158)(36,83,39,165)(49,103,132,152)(50,98,121,147)(51,105,122,154)(52,100,123,149)(53,107,124,156)(54,102,125,151)(55,97,126,146)(56,104,127,153)(57,99,128,148)(58,106,129,155)(59,101,130,150)(60,108,131,145)(61,171,136,96)(62,178,137,91)(63,173,138,86)(64,180,139,93)(65,175,140,88)(66,170,141,95)(67,177,142,90)(68,172,143,85)(69,179,144,92)(70,174,133,87)(71,169,134,94)(72,176,135,89)>;
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)(157,158,159,160,161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192), (1,82,7,76)(2,81,8,75)(3,80,9,74)(4,79,10,73)(5,78,11,84)(6,77,12,83)(13,47,19,41)(14,46,20,40)(15,45,21,39)(16,44,22,38)(17,43,23,37)(18,42,24,48)(25,181,31,187)(26,192,32,186)(27,191,33,185)(28,190,34,184)(29,189,35,183)(30,188,36,182)(49,62,55,68)(50,61,56,67)(51,72,57,66)(52,71,58,65)(53,70,59,64)(54,69,60,63)(85,152,91,146)(86,151,92,145)(87,150,93,156)(88,149,94,155)(89,148,95,154)(90,147,96,153)(97,172,103,178)(98,171,104,177)(99,170,105,176)(100,169,106,175)(101,180,107,174)(102,179,108,173)(109,167,115,161)(110,166,116,160)(111,165,117,159)(112,164,118,158)(113,163,119,157)(114,162,120,168)(121,136,127,142)(122,135,128,141)(123,134,129,140)(124,133,130,139)(125,144,131,138)(126,143,132,137), (1,148,112,99)(2,149,113,100)(3,150,114,101)(4,151,115,102)(5,152,116,103)(6,153,117,104)(7,154,118,105)(8,155,119,106)(9,156,120,107)(10,145,109,108)(11,146,110,97)(12,147,111,98)(13,125,192,54)(14,126,181,55)(15,127,182,56)(16,128,183,57)(17,129,184,58)(18,130,185,59)(19,131,186,60)(20,132,187,49)(21,121,188,50)(22,122,189,51)(23,123,190,52)(24,124,191,53)(25,62,40,137)(26,63,41,138)(27,64,42,139)(28,65,43,140)(29,66,44,141)(30,67,45,142)(31,68,46,143)(32,69,47,144)(33,70,48,133)(34,71,37,134)(35,72,38,135)(36,61,39,136)(73,86,167,173)(74,87,168,174)(75,88,157,175)(76,89,158,176)(77,90,159,177)(78,91,160,178)(79,92,161,179)(80,93,162,180)(81,94,163,169)(82,95,164,170)(83,96,165,171)(84,85,166,172), (1,16,112,183)(2,23,113,190)(3,18,114,185)(4,13,115,192)(5,20,116,187)(6,15,117,182)(7,22,118,189)(8,17,119,184)(9,24,120,191)(10,19,109,186)(11,14,110,181)(12,21,111,188)(25,78,40,160)(26,73,41,167)(27,80,42,162)(28,75,43,157)(29,82,44,164)(30,77,45,159)(31,84,46,166)(32,79,47,161)(33,74,48,168)(34,81,37,163)(35,76,38,158)(36,83,39,165)(49,103,132,152)(50,98,121,147)(51,105,122,154)(52,100,123,149)(53,107,124,156)(54,102,125,151)(55,97,126,146)(56,104,127,153)(57,99,128,148)(58,106,129,155)(59,101,130,150)(60,108,131,145)(61,171,136,96)(62,178,137,91)(63,173,138,86)(64,180,139,93)(65,175,140,88)(66,170,141,95)(67,177,142,90)(68,172,143,85)(69,179,144,92)(70,174,133,87)(71,169,134,94)(72,176,135,89) );
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),(157,158,159,160,161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192)], [(1,82,7,76),(2,81,8,75),(3,80,9,74),(4,79,10,73),(5,78,11,84),(6,77,12,83),(13,47,19,41),(14,46,20,40),(15,45,21,39),(16,44,22,38),(17,43,23,37),(18,42,24,48),(25,181,31,187),(26,192,32,186),(27,191,33,185),(28,190,34,184),(29,189,35,183),(30,188,36,182),(49,62,55,68),(50,61,56,67),(51,72,57,66),(52,71,58,65),(53,70,59,64),(54,69,60,63),(85,152,91,146),(86,151,92,145),(87,150,93,156),(88,149,94,155),(89,148,95,154),(90,147,96,153),(97,172,103,178),(98,171,104,177),(99,170,105,176),(100,169,106,175),(101,180,107,174),(102,179,108,173),(109,167,115,161),(110,166,116,160),(111,165,117,159),(112,164,118,158),(113,163,119,157),(114,162,120,168),(121,136,127,142),(122,135,128,141),(123,134,129,140),(124,133,130,139),(125,144,131,138),(126,143,132,137)], [(1,148,112,99),(2,149,113,100),(3,150,114,101),(4,151,115,102),(5,152,116,103),(6,153,117,104),(7,154,118,105),(8,155,119,106),(9,156,120,107),(10,145,109,108),(11,146,110,97),(12,147,111,98),(13,125,192,54),(14,126,181,55),(15,127,182,56),(16,128,183,57),(17,129,184,58),(18,130,185,59),(19,131,186,60),(20,132,187,49),(21,121,188,50),(22,122,189,51),(23,123,190,52),(24,124,191,53),(25,62,40,137),(26,63,41,138),(27,64,42,139),(28,65,43,140),(29,66,44,141),(30,67,45,142),(31,68,46,143),(32,69,47,144),(33,70,48,133),(34,71,37,134),(35,72,38,135),(36,61,39,136),(73,86,167,173),(74,87,168,174),(75,88,157,175),(76,89,158,176),(77,90,159,177),(78,91,160,178),(79,92,161,179),(80,93,162,180),(81,94,163,169),(82,95,164,170),(83,96,165,171),(84,85,166,172)], [(1,16,112,183),(2,23,113,190),(3,18,114,185),(4,13,115,192),(5,20,116,187),(6,15,117,182),(7,22,118,189),(8,17,119,184),(9,24,120,191),(10,19,109,186),(11,14,110,181),(12,21,111,188),(25,78,40,160),(26,73,41,167),(27,80,42,162),(28,75,43,157),(29,82,44,164),(30,77,45,159),(31,84,46,166),(32,79,47,161),(33,74,48,168),(34,81,37,163),(35,76,38,158),(36,83,39,165),(49,103,132,152),(50,98,121,147),(51,105,122,154),(52,100,123,149),(53,107,124,156),(54,102,125,151),(55,97,126,146),(56,104,127,153),(57,99,128,148),(58,106,129,155),(59,101,130,150),(60,108,131,145),(61,171,136,96),(62,178,137,91),(63,173,138,86),(64,180,139,93),(65,175,140,88),(66,170,141,95),(67,177,142,90),(68,172,143,85),(69,179,144,92),(70,174,133,87),(71,169,134,94),(72,176,135,89)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 11 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 5 | 0 | 0 | 0 | 0 |
| 10 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 8 | 1 | 0 | 0 | 0 | 0 |
| 2 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 0 | 8 | 0 |
G:=sub<GL(6,GF(13))| [1,1,0,0,0,0,11,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,10,0,0,0,0,5,12,0,0,0,0,0,0,12,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,0],[8,2,0,0,0,0,1,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,8,0] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | ··· | 4Q | 4R | 4S | 4T | 4U | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | C4○D4 | 2- (1+4) | D4⋊2S3 | S3×Q8 | Q8.15D6 |
| kernel | Dic6⋊8Q8 | C4×Dic6 | Dic6⋊C4 | Dic3.Q8 | C12.3Q8 | Dic3⋊Q8 | Q8×Dic3 | C3×C4⋊Q8 | C4⋊Q8 | Dic6 | C42 | C4⋊C4 | C2×Q8 | C12 | C6 | C4 | C4 | C2 |
| # reps | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 1 | 1 | 4 | 1 | 4 | 2 | 4 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
Dic_6\rtimes_8Q_8
% in TeX
G:=Group("Dic6:8Q8"); // GroupNames label
G:=SmallGroup(192,1280);
// by ID
G=gap.SmallGroup(192,1280);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,120,219,268,1571,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^4=1,b^2=a^6,d^2=c^2,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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