metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.3(C4×Q8), C4⋊Dic3⋊8C4, C6.45(C4×D4), (C2×C12).34Q8, C2.6(C4×Dic6), (C2×C4).21Dic6, C22.55(S3×D4), (C22×C4).26D6, C6.3(C22⋊Q8), C6.9(C42.C2), C2.4(Dic3⋊5D4), (C2×Dic3).128D4, C2.2(C23.9D6), C2.1(C4.Dic6), C22.14(C2×Dic6), C6.17(C42⋊2C2), C6.19(C42⋊C2), C22.31(C4○D12), C2.C42.11S3, C6.C42.23C2, (C22×C6).279C23, (C22×C12).43C22, C23.261(C22×S3), C6.3(C22.D4), C22.33(D4⋊2S3), C2.3(Dic3.D4), C3⋊3(C23.63C23), C22.14(Q8⋊3S3), C2.8(C23.16D6), (C22×Dic3).5C22, (C2×C4).25(C4×S3), C22.86(S3×C2×C4), (C2×C6).19(C2×Q8), (C2×C12).33(C2×C4), (C2×C6).192(C2×D4), (C2×C4⋊Dic3).4C2, C2.2(C4⋊C4⋊S3), (C2×C4×Dic3).27C2, (C2×Dic3⋊C4).5C2, (C2×C6).45(C22×C4), (C2×C6).180(C4○D4), (C2×Dic3).43(C2×C4), (C3×C2.C42).17C2, SmallGroup(192,213)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2.(C4×Dic6)
G = < a,b,c,d | a2=b4=c12=1, d2=ac6, cbc-1=ab=ba, ac=ca, ad=da, bd=db, dcd-1=c-1 >
Subgroups: 352 in 154 conjugacy classes, 69 normal (51 characteristic)
C1, C2, C3, C4, C22, C6, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C42, C4⋊C4, C22×C4, C22×C4, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C22×Dic3, C22×C12, C23.63C23, C6.C42, C3×C2.C42, C2×C4×Dic3, C2×Dic3⋊C4, C2×C4⋊Dic3, C2.(C4×Dic6)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, C22×S3, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C42⋊2C2, C2×Dic6, S3×C2×C4, C4○D12, S3×D4, D4⋊2S3, Q8⋊3S3, C23.63C23, C4×Dic6, C23.16D6, Dic3.D4, C23.9D6, C4.Dic6, Dic3⋊5D4, C4⋊C4⋊S3, C2.(C4×Dic6)
►Regular action on 192 points
Generators in S192
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 49)(25 115)(26 116)(27 117)(28 118)(29 119)(30 120)(31 109)(32 110)(33 111)(34 112)(35 113)(36 114)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(71 73)(72 74)(85 192)(86 181)(87 182)(88 183)(89 184)(90 185)(91 186)(92 187)(93 188)(94 189)(95 190)(96 191)(97 140)(98 141)(99 142)(100 143)(101 144)(102 133)(103 134)(104 135)(105 136)(106 137)(107 138)(108 139)(121 165)(122 166)(123 167)(124 168)(125 157)(126 158)(127 159)(128 160)(129 161)(130 162)(131 163)(132 164)(145 180)(146 169)(147 170)(148 171)(149 172)(150 173)(151 174)(152 175)(153 176)(154 177)(155 178)(156 179)
(1 190 18 125)(2 96 19 158)(3 192 20 127)(4 86 21 160)(5 182 22 129)(6 88 23 162)(7 184 24 131)(8 90 13 164)(9 186 14 121)(10 92 15 166)(11 188 16 123)(12 94 17 168)(25 176 141 63)(26 154 142 78)(27 178 143 65)(28 156 144 80)(29 180 133 67)(30 146 134 82)(31 170 135 69)(32 148 136 84)(33 172 137 71)(34 150 138 74)(35 174 139 61)(36 152 140 76)(37 95 55 157)(38 191 56 126)(39 85 57 159)(40 181 58 128)(41 87 59 161)(42 183 60 130)(43 89 49 163)(44 185 50 132)(45 91 51 165)(46 187 52 122)(47 93 53 167)(48 189 54 124)(62 114 175 97)(64 116 177 99)(66 118 179 101)(68 120 169 103)(70 110 171 105)(72 112 173 107)(73 111 149 106)(75 113 151 108)(77 115 153 98)(79 117 155 100)(81 119 145 102)(83 109 147 104)
(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)
(1 119 43 35)(2 118 44 34)(3 117 45 33)(4 116 46 32)(5 115 47 31)(6 114 48 30)(7 113 37 29)(8 112 38 28)(9 111 39 27)(10 110 40 26)(11 109 41 25)(12 120 42 36)(13 107 56 144)(14 106 57 143)(15 105 58 142)(16 104 59 141)(17 103 60 140)(18 102 49 139)(19 101 50 138)(20 100 51 137)(21 99 52 136)(22 98 53 135)(23 97 54 134)(24 108 55 133)(61 125 81 163)(62 124 82 162)(63 123 83 161)(64 122 84 160)(65 121 73 159)(66 132 74 158)(67 131 75 157)(68 130 76 168)(69 129 77 167)(70 128 78 166)(71 127 79 165)(72 126 80 164)(85 178 186 149)(86 177 187 148)(87 176 188 147)(88 175 189 146)(89 174 190 145)(90 173 191 156)(91 172 192 155)(92 171 181 154)(93 170 182 153)(94 169 183 152)(95 180 184 151)(96 179 185 150)
G:=sub<Sym(192)| (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,49)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(31,109)(32,110)(33,111)(34,112)(35,113)(36,114)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(71,73)(72,74)(85,192)(86,181)(87,182)(88,183)(89,184)(90,185)(91,186)(92,187)(93,188)(94,189)(95,190)(96,191)(97,140)(98,141)(99,142)(100,143)(101,144)(102,133)(103,134)(104,135)(105,136)(106,137)(107,138)(108,139)(121,165)(122,166)(123,167)(124,168)(125,157)(126,158)(127,159)(128,160)(129,161)(130,162)(131,163)(132,164)(145,180)(146,169)(147,170)(148,171)(149,172)(150,173)(151,174)(152,175)(153,176)(154,177)(155,178)(156,179), (1,190,18,125)(2,96,19,158)(3,192,20,127)(4,86,21,160)(5,182,22,129)(6,88,23,162)(7,184,24,131)(8,90,13,164)(9,186,14,121)(10,92,15,166)(11,188,16,123)(12,94,17,168)(25,176,141,63)(26,154,142,78)(27,178,143,65)(28,156,144,80)(29,180,133,67)(30,146,134,82)(31,170,135,69)(32,148,136,84)(33,172,137,71)(34,150,138,74)(35,174,139,61)(36,152,140,76)(37,95,55,157)(38,191,56,126)(39,85,57,159)(40,181,58,128)(41,87,59,161)(42,183,60,130)(43,89,49,163)(44,185,50,132)(45,91,51,165)(46,187,52,122)(47,93,53,167)(48,189,54,124)(62,114,175,97)(64,116,177,99)(66,118,179,101)(68,120,169,103)(70,110,171,105)(72,112,173,107)(73,111,149,106)(75,113,151,108)(77,115,153,98)(79,117,155,100)(81,119,145,102)(83,109,147,104), (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), (1,119,43,35)(2,118,44,34)(3,117,45,33)(4,116,46,32)(5,115,47,31)(6,114,48,30)(7,113,37,29)(8,112,38,28)(9,111,39,27)(10,110,40,26)(11,109,41,25)(12,120,42,36)(13,107,56,144)(14,106,57,143)(15,105,58,142)(16,104,59,141)(17,103,60,140)(18,102,49,139)(19,101,50,138)(20,100,51,137)(21,99,52,136)(22,98,53,135)(23,97,54,134)(24,108,55,133)(61,125,81,163)(62,124,82,162)(63,123,83,161)(64,122,84,160)(65,121,73,159)(66,132,74,158)(67,131,75,157)(68,130,76,168)(69,129,77,167)(70,128,78,166)(71,127,79,165)(72,126,80,164)(85,178,186,149)(86,177,187,148)(87,176,188,147)(88,175,189,146)(89,174,190,145)(90,173,191,156)(91,172,192,155)(92,171,181,154)(93,170,182,153)(94,169,183,152)(95,180,184,151)(96,179,185,150)>;
G:=Group( (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,49)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(31,109)(32,110)(33,111)(34,112)(35,113)(36,114)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(71,73)(72,74)(85,192)(86,181)(87,182)(88,183)(89,184)(90,185)(91,186)(92,187)(93,188)(94,189)(95,190)(96,191)(97,140)(98,141)(99,142)(100,143)(101,144)(102,133)(103,134)(104,135)(105,136)(106,137)(107,138)(108,139)(121,165)(122,166)(123,167)(124,168)(125,157)(126,158)(127,159)(128,160)(129,161)(130,162)(131,163)(132,164)(145,180)(146,169)(147,170)(148,171)(149,172)(150,173)(151,174)(152,175)(153,176)(154,177)(155,178)(156,179), (1,190,18,125)(2,96,19,158)(3,192,20,127)(4,86,21,160)(5,182,22,129)(6,88,23,162)(7,184,24,131)(8,90,13,164)(9,186,14,121)(10,92,15,166)(11,188,16,123)(12,94,17,168)(25,176,141,63)(26,154,142,78)(27,178,143,65)(28,156,144,80)(29,180,133,67)(30,146,134,82)(31,170,135,69)(32,148,136,84)(33,172,137,71)(34,150,138,74)(35,174,139,61)(36,152,140,76)(37,95,55,157)(38,191,56,126)(39,85,57,159)(40,181,58,128)(41,87,59,161)(42,183,60,130)(43,89,49,163)(44,185,50,132)(45,91,51,165)(46,187,52,122)(47,93,53,167)(48,189,54,124)(62,114,175,97)(64,116,177,99)(66,118,179,101)(68,120,169,103)(70,110,171,105)(72,112,173,107)(73,111,149,106)(75,113,151,108)(77,115,153,98)(79,117,155,100)(81,119,145,102)(83,109,147,104), (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), (1,119,43,35)(2,118,44,34)(3,117,45,33)(4,116,46,32)(5,115,47,31)(6,114,48,30)(7,113,37,29)(8,112,38,28)(9,111,39,27)(10,110,40,26)(11,109,41,25)(12,120,42,36)(13,107,56,144)(14,106,57,143)(15,105,58,142)(16,104,59,141)(17,103,60,140)(18,102,49,139)(19,101,50,138)(20,100,51,137)(21,99,52,136)(22,98,53,135)(23,97,54,134)(24,108,55,133)(61,125,81,163)(62,124,82,162)(63,123,83,161)(64,122,84,160)(65,121,73,159)(66,132,74,158)(67,131,75,157)(68,130,76,168)(69,129,77,167)(70,128,78,166)(71,127,79,165)(72,126,80,164)(85,178,186,149)(86,177,187,148)(87,176,188,147)(88,175,189,146)(89,174,190,145)(90,173,191,156)(91,172,192,155)(92,171,181,154)(93,170,182,153)(94,169,183,152)(95,180,184,151)(96,179,185,150) );
G=PermutationGroup([[(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,43),(8,44),(9,45),(10,46),(11,47),(12,48),(13,50),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,49),(25,115),(26,116),(27,117),(28,118),(29,119),(30,120),(31,109),(32,110),(33,111),(34,112),(35,113),(36,114),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(71,73),(72,74),(85,192),(86,181),(87,182),(88,183),(89,184),(90,185),(91,186),(92,187),(93,188),(94,189),(95,190),(96,191),(97,140),(98,141),(99,142),(100,143),(101,144),(102,133),(103,134),(104,135),(105,136),(106,137),(107,138),(108,139),(121,165),(122,166),(123,167),(124,168),(125,157),(126,158),(127,159),(128,160),(129,161),(130,162),(131,163),(132,164),(145,180),(146,169),(147,170),(148,171),(149,172),(150,173),(151,174),(152,175),(153,176),(154,177),(155,178),(156,179)], [(1,190,18,125),(2,96,19,158),(3,192,20,127),(4,86,21,160),(5,182,22,129),(6,88,23,162),(7,184,24,131),(8,90,13,164),(9,186,14,121),(10,92,15,166),(11,188,16,123),(12,94,17,168),(25,176,141,63),(26,154,142,78),(27,178,143,65),(28,156,144,80),(29,180,133,67),(30,146,134,82),(31,170,135,69),(32,148,136,84),(33,172,137,71),(34,150,138,74),(35,174,139,61),(36,152,140,76),(37,95,55,157),(38,191,56,126),(39,85,57,159),(40,181,58,128),(41,87,59,161),(42,183,60,130),(43,89,49,163),(44,185,50,132),(45,91,51,165),(46,187,52,122),(47,93,53,167),(48,189,54,124),(62,114,175,97),(64,116,177,99),(66,118,179,101),(68,120,169,103),(70,110,171,105),(72,112,173,107),(73,111,149,106),(75,113,151,108),(77,115,153,98),(79,117,155,100),(81,119,145,102),(83,109,147,104)], [(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)], [(1,119,43,35),(2,118,44,34),(3,117,45,33),(4,116,46,32),(5,115,47,31),(6,114,48,30),(7,113,37,29),(8,112,38,28),(9,111,39,27),(10,110,40,26),(11,109,41,25),(12,120,42,36),(13,107,56,144),(14,106,57,143),(15,105,58,142),(16,104,59,141),(17,103,60,140),(18,102,49,139),(19,101,50,138),(20,100,51,137),(21,99,52,136),(22,98,53,135),(23,97,54,134),(24,108,55,133),(61,125,81,163),(62,124,82,162),(63,123,83,161),(64,122,84,160),(65,121,73,159),(66,132,74,158),(67,131,75,157),(68,130,76,168),(69,129,77,167),(70,128,78,166),(71,127,79,165),(72,126,80,164),(85,178,186,149),(86,177,187,148),(87,176,188,147),(88,175,189,146),(89,174,190,145),(90,173,191,156),(91,172,192,155),(92,171,181,154),(93,170,182,153),(94,169,183,152),(95,180,184,151),(96,179,185,150)]])
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 4S | 4T | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | D6 | C4○D4 | Dic6 | C4×S3 | C4○D12 | S3×D4 | D4⋊2S3 | Q8⋊3S3 |
| kernel | C2.(C4×Dic6) | C6.C42 | C3×C2.C42 | C2×C4×Dic3 | C2×Dic3⋊C4 | C2×C4⋊Dic3 | C4⋊Dic3 | C2.C42 | C2×Dic3 | C2×C12 | C22×C4 | C2×C6 | C2×C4 | C2×C4 | C22 | C22 | C22 | C22 |
| # reps | 1 | 3 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 3 | 8 | 4 | 4 | 4 | 1 | 2 | 1 |
Matrix representation of C2.(C4×Dic6) ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 10 |
| 0 | 0 | 0 | 0 | 3 | 3 |
| 9 | 10 | 0 | 0 | 0 | 0 |
| 10 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 9 |
| 0 | 0 | 0 | 0 | 11 | 11 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,5,0,0,0,0,0,0,0,6,3,0,0,0,0,10,3],[9,10,0,0,0,0,10,4,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,2,11,0,0,0,0,9,11] >;
C2.(C4×Dic6) in GAP, Magma, Sage, TeX
C_2.(C_4\times {\rm Dic}_6) % in TeX
G:=Group("C2.(C4xDic6)"); // GroupNames label
G:=SmallGroup(192,213);
// by ID
G=gap.SmallGroup(192,213);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,701,344,387,58,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^12=1,d^2=a*c^6,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations