direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4×Dic6, C42.271D6, C6⋊1(C4×Q8), C12⋊6(C2×Q8), (C2×C12)⋊14Q8, C6.1(C23×C4), C6.1(C22×Q8), (C2×C6).11C24, (C2×C42).19S3, (C22×C4).481D6, C2.1(C22×Dic6), (C4×C12).332C22, (C2×C12).873C23, C12.117(C22×C4), C22.11(S3×C23), Dic3.1(C22×C4), C22.33(C2×Dic6), C22.65(C4○D12), C4⋊Dic3.394C22, C23.319(C22×S3), (C22×C6).373C23, (C22×Dic6).20C2, Dic3⋊C4.173C22, (C22×C12).500C22, (C2×Dic3).171C23, (C2×Dic6).313C22, (C4×Dic3).287C22, (C22×Dic3).199C22, C3⋊1(C2×C4×Q8), C4.75(S3×C2×C4), (C2×C4×C12).22C2, C6.1(C2×C4○D4), C2.4(S3×C22×C4), C2.1(C2×C4○D12), C22.66(S3×C2×C4), (C2×C6).45(C2×Q8), (C2×C4).118(C4×S3), (C2×C4×Dic3).39C2, (C2×C12).230(C2×C4), (C2×C6).93(C4○D4), (C2×C4⋊Dic3).47C2, (C2×Dic3⋊C4).36C2, (C2×C6).144(C22×C4), (C2×C4).815(C22×S3), (C2×Dic3).69(C2×C4), SmallGroup(192,1026)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 536 in 298 conjugacy classes, 183 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×8], C4 [×14], C22, C22 [×6], C6 [×3], C6 [×4], C2×C4 [×14], C2×C4 [×22], Q8 [×16], C23, Dic3 [×8], Dic3 [×4], C12 [×8], C12 [×2], C2×C6, C2×C6 [×6], C42 [×4], C42 [×8], C4⋊C4 [×12], C22×C4 [×3], C22×C4 [×4], C2×Q8 [×12], Dic6 [×16], C2×Dic3 [×16], C2×Dic3 [×4], C2×C12 [×14], C2×C12 [×2], C22×C6, C2×C42, C2×C42 [×2], C2×C4⋊C4 [×3], C4×Q8 [×8], C22×Q8, C4×Dic3 [×8], Dic3⋊C4 [×8], C4⋊Dic3 [×4], C4×C12 [×4], C2×Dic6 [×12], C22×Dic3 [×4], C22×C12 [×3], C2×C4×Q8, C4×Dic6 [×8], C2×C4×Dic3 [×2], C2×Dic3⋊C4 [×2], C2×C4⋊Dic3, C2×C4×C12, C22×Dic6, C2×C4×Dic6
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], Q8 [×4], C23 [×15], D6 [×7], C22×C4 [×14], C2×Q8 [×6], C4○D4 [×2], C24, Dic6 [×4], C4×S3 [×4], C22×S3 [×7], C4×Q8 [×4], C23×C4, C22×Q8, C2×C4○D4, C2×Dic6 [×6], S3×C2×C4 [×6], C4○D12 [×2], S3×C23, C2×C4×Q8, C4×Dic6 [×4], C22×Dic6, S3×C22×C4, C2×C4○D12, C2×C4×Dic6
Generators and relations
G = < a,b,c,d | a2=b4=c12=1, d2=c6, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
►Regular action on 192 points
Generators in S192
(1 20)(2 21)(3 22)(4 23)(5 24)(6 13)(7 14)(8 15)(9 16)(10 17)(11 18)(12 19)(25 56)(26 57)(27 58)(28 59)(29 60)(30 49)(31 50)(32 51)(33 52)(34 53)(35 54)(36 55)(37 151)(38 152)(39 153)(40 154)(41 155)(42 156)(43 145)(44 146)(45 147)(46 148)(47 149)(48 150)(61 79)(62 80)(63 81)(64 82)(65 83)(66 84)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)(85 184)(86 185)(87 186)(88 187)(89 188)(90 189)(91 190)(92 191)(93 192)(94 181)(95 182)(96 183)(97 135)(98 136)(99 137)(100 138)(101 139)(102 140)(103 141)(104 142)(105 143)(106 144)(107 133)(108 134)(109 162)(110 163)(111 164)(112 165)(113 166)(114 167)(115 168)(116 157)(117 158)(118 159)(119 160)(120 161)(121 172)(122 173)(123 174)(124 175)(125 176)(126 177)(127 178)(128 179)(129 180)(130 169)(131 170)(132 171)
(1 173 140 47)(2 174 141 48)(3 175 142 37)(4 176 143 38)(5 177 144 39)(6 178 133 40)(7 179 134 41)(8 180 135 42)(9 169 136 43)(10 170 137 44)(11 171 138 45)(12 172 139 46)(13 127 107 154)(14 128 108 155)(15 129 97 156)(16 130 98 145)(17 131 99 146)(18 132 100 147)(19 121 101 148)(20 122 102 149)(21 123 103 150)(22 124 104 151)(23 125 105 152)(24 126 106 153)(25 63 158 191)(26 64 159 192)(27 65 160 181)(28 66 161 182)(29 67 162 183)(30 68 163 184)(31 69 164 185)(32 70 165 186)(33 71 166 187)(34 72 167 188)(35 61 168 189)(36 62 157 190)(49 74 110 85)(50 75 111 86)(51 76 112 87)(52 77 113 88)(53 78 114 89)(54 79 115 90)(55 80 116 91)(56 81 117 92)(57 82 118 93)(58 83 119 94)(59 84 120 95)(60 73 109 96)
(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 54 7 60)(2 53 8 59)(3 52 9 58)(4 51 10 57)(5 50 11 56)(6 49 12 55)(13 30 19 36)(14 29 20 35)(15 28 21 34)(16 27 22 33)(17 26 23 32)(18 25 24 31)(37 88 43 94)(38 87 44 93)(39 86 45 92)(40 85 46 91)(41 96 47 90)(42 95 48 89)(61 128 67 122)(62 127 68 121)(63 126 69 132)(64 125 70 131)(65 124 71 130)(66 123 72 129)(73 173 79 179)(74 172 80 178)(75 171 81 177)(76 170 82 176)(77 169 83 175)(78 180 84 174)(97 161 103 167)(98 160 104 166)(99 159 105 165)(100 158 106 164)(101 157 107 163)(102 168 108 162)(109 140 115 134)(110 139 116 133)(111 138 117 144)(112 137 118 143)(113 136 119 142)(114 135 120 141)(145 181 151 187)(146 192 152 186)(147 191 153 185)(148 190 154 184)(149 189 155 183)(150 188 156 182)
G:=sub<Sym(192)| (1,20)(2,21)(3,22)(4,23)(5,24)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19)(25,56)(26,57)(27,58)(28,59)(29,60)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,151)(38,152)(39,153)(40,154)(41,155)(42,156)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(61,79)(62,80)(63,81)(64,82)(65,83)(66,84)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78)(85,184)(86,185)(87,186)(88,187)(89,188)(90,189)(91,190)(92,191)(93,192)(94,181)(95,182)(96,183)(97,135)(98,136)(99,137)(100,138)(101,139)(102,140)(103,141)(104,142)(105,143)(106,144)(107,133)(108,134)(109,162)(110,163)(111,164)(112,165)(113,166)(114,167)(115,168)(116,157)(117,158)(118,159)(119,160)(120,161)(121,172)(122,173)(123,174)(124,175)(125,176)(126,177)(127,178)(128,179)(129,180)(130,169)(131,170)(132,171), (1,173,140,47)(2,174,141,48)(3,175,142,37)(4,176,143,38)(5,177,144,39)(6,178,133,40)(7,179,134,41)(8,180,135,42)(9,169,136,43)(10,170,137,44)(11,171,138,45)(12,172,139,46)(13,127,107,154)(14,128,108,155)(15,129,97,156)(16,130,98,145)(17,131,99,146)(18,132,100,147)(19,121,101,148)(20,122,102,149)(21,123,103,150)(22,124,104,151)(23,125,105,152)(24,126,106,153)(25,63,158,191)(26,64,159,192)(27,65,160,181)(28,66,161,182)(29,67,162,183)(30,68,163,184)(31,69,164,185)(32,70,165,186)(33,71,166,187)(34,72,167,188)(35,61,168,189)(36,62,157,190)(49,74,110,85)(50,75,111,86)(51,76,112,87)(52,77,113,88)(53,78,114,89)(54,79,115,90)(55,80,116,91)(56,81,117,92)(57,82,118,93)(58,83,119,94)(59,84,120,95)(60,73,109,96), (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), (1,54,7,60)(2,53,8,59)(3,52,9,58)(4,51,10,57)(5,50,11,56)(6,49,12,55)(13,30,19,36)(14,29,20,35)(15,28,21,34)(16,27,22,33)(17,26,23,32)(18,25,24,31)(37,88,43,94)(38,87,44,93)(39,86,45,92)(40,85,46,91)(41,96,47,90)(42,95,48,89)(61,128,67,122)(62,127,68,121)(63,126,69,132)(64,125,70,131)(65,124,71,130)(66,123,72,129)(73,173,79,179)(74,172,80,178)(75,171,81,177)(76,170,82,176)(77,169,83,175)(78,180,84,174)(97,161,103,167)(98,160,104,166)(99,159,105,165)(100,158,106,164)(101,157,107,163)(102,168,108,162)(109,140,115,134)(110,139,116,133)(111,138,117,144)(112,137,118,143)(113,136,119,142)(114,135,120,141)(145,181,151,187)(146,192,152,186)(147,191,153,185)(148,190,154,184)(149,189,155,183)(150,188,156,182)>;
G:=Group( (1,20)(2,21)(3,22)(4,23)(5,24)(6,13)(7,14)(8,15)(9,16)(10,17)(11,18)(12,19)(25,56)(26,57)(27,58)(28,59)(29,60)(30,49)(31,50)(32,51)(33,52)(34,53)(35,54)(36,55)(37,151)(38,152)(39,153)(40,154)(41,155)(42,156)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(61,79)(62,80)(63,81)(64,82)(65,83)(66,84)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78)(85,184)(86,185)(87,186)(88,187)(89,188)(90,189)(91,190)(92,191)(93,192)(94,181)(95,182)(96,183)(97,135)(98,136)(99,137)(100,138)(101,139)(102,140)(103,141)(104,142)(105,143)(106,144)(107,133)(108,134)(109,162)(110,163)(111,164)(112,165)(113,166)(114,167)(115,168)(116,157)(117,158)(118,159)(119,160)(120,161)(121,172)(122,173)(123,174)(124,175)(125,176)(126,177)(127,178)(128,179)(129,180)(130,169)(131,170)(132,171), (1,173,140,47)(2,174,141,48)(3,175,142,37)(4,176,143,38)(5,177,144,39)(6,178,133,40)(7,179,134,41)(8,180,135,42)(9,169,136,43)(10,170,137,44)(11,171,138,45)(12,172,139,46)(13,127,107,154)(14,128,108,155)(15,129,97,156)(16,130,98,145)(17,131,99,146)(18,132,100,147)(19,121,101,148)(20,122,102,149)(21,123,103,150)(22,124,104,151)(23,125,105,152)(24,126,106,153)(25,63,158,191)(26,64,159,192)(27,65,160,181)(28,66,161,182)(29,67,162,183)(30,68,163,184)(31,69,164,185)(32,70,165,186)(33,71,166,187)(34,72,167,188)(35,61,168,189)(36,62,157,190)(49,74,110,85)(50,75,111,86)(51,76,112,87)(52,77,113,88)(53,78,114,89)(54,79,115,90)(55,80,116,91)(56,81,117,92)(57,82,118,93)(58,83,119,94)(59,84,120,95)(60,73,109,96), (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), (1,54,7,60)(2,53,8,59)(3,52,9,58)(4,51,10,57)(5,50,11,56)(6,49,12,55)(13,30,19,36)(14,29,20,35)(15,28,21,34)(16,27,22,33)(17,26,23,32)(18,25,24,31)(37,88,43,94)(38,87,44,93)(39,86,45,92)(40,85,46,91)(41,96,47,90)(42,95,48,89)(61,128,67,122)(62,127,68,121)(63,126,69,132)(64,125,70,131)(65,124,71,130)(66,123,72,129)(73,173,79,179)(74,172,80,178)(75,171,81,177)(76,170,82,176)(77,169,83,175)(78,180,84,174)(97,161,103,167)(98,160,104,166)(99,159,105,165)(100,158,106,164)(101,157,107,163)(102,168,108,162)(109,140,115,134)(110,139,116,133)(111,138,117,144)(112,137,118,143)(113,136,119,142)(114,135,120,141)(145,181,151,187)(146,192,152,186)(147,191,153,185)(148,190,154,184)(149,189,155,183)(150,188,156,182) );
G=PermutationGroup([(1,20),(2,21),(3,22),(4,23),(5,24),(6,13),(7,14),(8,15),(9,16),(10,17),(11,18),(12,19),(25,56),(26,57),(27,58),(28,59),(29,60),(30,49),(31,50),(32,51),(33,52),(34,53),(35,54),(36,55),(37,151),(38,152),(39,153),(40,154),(41,155),(42,156),(43,145),(44,146),(45,147),(46,148),(47,149),(48,150),(61,79),(62,80),(63,81),(64,82),(65,83),(66,84),(67,73),(68,74),(69,75),(70,76),(71,77),(72,78),(85,184),(86,185),(87,186),(88,187),(89,188),(90,189),(91,190),(92,191),(93,192),(94,181),(95,182),(96,183),(97,135),(98,136),(99,137),(100,138),(101,139),(102,140),(103,141),(104,142),(105,143),(106,144),(107,133),(108,134),(109,162),(110,163),(111,164),(112,165),(113,166),(114,167),(115,168),(116,157),(117,158),(118,159),(119,160),(120,161),(121,172),(122,173),(123,174),(124,175),(125,176),(126,177),(127,178),(128,179),(129,180),(130,169),(131,170),(132,171)], [(1,173,140,47),(2,174,141,48),(3,175,142,37),(4,176,143,38),(5,177,144,39),(6,178,133,40),(7,179,134,41),(8,180,135,42),(9,169,136,43),(10,170,137,44),(11,171,138,45),(12,172,139,46),(13,127,107,154),(14,128,108,155),(15,129,97,156),(16,130,98,145),(17,131,99,146),(18,132,100,147),(19,121,101,148),(20,122,102,149),(21,123,103,150),(22,124,104,151),(23,125,105,152),(24,126,106,153),(25,63,158,191),(26,64,159,192),(27,65,160,181),(28,66,161,182),(29,67,162,183),(30,68,163,184),(31,69,164,185),(32,70,165,186),(33,71,166,187),(34,72,167,188),(35,61,168,189),(36,62,157,190),(49,74,110,85),(50,75,111,86),(51,76,112,87),(52,77,113,88),(53,78,114,89),(54,79,115,90),(55,80,116,91),(56,81,117,92),(57,82,118,93),(58,83,119,94),(59,84,120,95),(60,73,109,96)], [(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)], [(1,54,7,60),(2,53,8,59),(3,52,9,58),(4,51,10,57),(5,50,11,56),(6,49,12,55),(13,30,19,36),(14,29,20,35),(15,28,21,34),(16,27,22,33),(17,26,23,32),(18,25,24,31),(37,88,43,94),(38,87,44,93),(39,86,45,92),(40,85,46,91),(41,96,47,90),(42,95,48,89),(61,128,67,122),(62,127,68,121),(63,126,69,132),(64,125,70,131),(65,124,71,130),(66,123,72,129),(73,173,79,179),(74,172,80,178),(75,171,81,177),(76,170,82,176),(77,169,83,175),(78,180,84,174),(97,161,103,167),(98,160,104,166),(99,159,105,165),(100,158,106,164),(101,157,107,163),(102,168,108,162),(109,140,115,134),(110,139,116,133),(111,138,117,144),(112,137,118,143),(113,136,119,142),(114,135,120,141),(145,181,151,187),(146,192,152,186),(147,191,153,185),(148,190,154,184),(149,189,155,183),(150,188,156,182)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 3 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 9 | 0 | 0 |
| 0 | 0 | 9 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,9,0,0,0,0,0,0,3],[4,3,0,0,0,0,3,9,0,0,0,0,0,0,3,9,0,0,0,0,9,10,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;
72 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4AF | 6A | ··· | 6G | 12A | ··· | 12X |
| order | 1 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | Q8 | D6 | D6 | C4○D4 | Dic6 | C4×S3 | C4○D12 |
| kernel | C2×C4×Dic6 | C4×Dic6 | C2×C4×Dic3 | C2×Dic3⋊C4 | C2×C4⋊Dic3 | C2×C4×C12 | C22×Dic6 | C2×Dic6 | C2×C42 | C2×C12 | C42 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 16 | 1 | 4 | 4 | 3 | 4 | 8 | 8 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_4\times Dic_6
% in TeX
G:=Group("C2xC4xDic6"); // GroupNames label
G:=SmallGroup(192,1026);
// by ID
G=gap.SmallGroup(192,1026);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,184,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^12=1,d^2=c^6,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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