direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×S3×Dic7, D6.11D14, C42.18C23, Dic21⋊8C22, C14⋊3(C4×S3), C42⋊2(C2×C4), (S3×C14)⋊2C4, C21⋊3(C22×C4), C6⋊1(C2×Dic7), (C6×Dic7)⋊3C2, (C2×C6).13D14, (C2×C14).14D6, C3⋊1(C22×Dic7), (C2×Dic21)⋊9C2, (C22×S3).2D7, C22.12(S3×D7), C6.18(C22×D7), C14.18(C22×S3), (C2×C42).12C22, (C3×Dic7)⋊5C22, (S3×C14).11C22, C7⋊4(S3×C2×C4), C2.3(C2×S3×D7), (S3×C7)⋊2(C2×C4), (S3×C2×C14).2C2, SmallGroup(336,154)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — C2×S3×Dic7 |
Generators and relations for C2×S3×Dic7
G = < a,b,c,d,e | a2=b3=c2=d14=1, e2=d7, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 412 in 108 conjugacy classes, 56 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C7, C2×C4, C23, Dic3, C12, D6, C2×C6, C14, C14, C14, C22×C4, C21, C4×S3, C2×Dic3, C2×C12, C22×S3, Dic7, Dic7, C2×C14, C2×C14, S3×C7, C42, C42, S3×C2×C4, C2×Dic7, C2×Dic7, C22×C14, C3×Dic7, Dic21, S3×C14, C2×C42, C22×Dic7, S3×Dic7, C6×Dic7, C2×Dic21, S3×C2×C14, C2×S3×Dic7
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, D7, C22×C4, C4×S3, C22×S3, Dic7, D14, S3×C2×C4, C2×Dic7, C22×D7, S3×D7, C22×Dic7, S3×Dic7, C2×S3×D7, C2×S3×Dic7
►On 168 points
Generators in S168
(1 152)(2 153)(3 154)(4 141)(5 142)(6 143)(7 144)(8 145)(9 146)(10 147)(11 148)(12 149)(13 150)(14 151)(15 140)(16 127)(17 128)(18 129)(19 130)(20 131)(21 132)(22 133)(23 134)(24 135)(25 136)(26 137)(27 138)(28 139)(29 66)(30 67)(31 68)(32 69)(33 70)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)(41 64)(42 65)(43 93)(44 94)(45 95)(46 96)(47 97)(48 98)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)(55 91)(56 92)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 99)(84 100)(113 159)(114 160)(115 161)(116 162)(117 163)(118 164)(119 165)(120 166)(121 167)(122 168)(123 155)(124 156)(125 157)(126 158)
(1 33 166)(2 34 167)(3 35 168)(4 36 155)(5 37 156)(6 38 157)(7 39 158)(8 40 159)(9 41 160)(10 42 161)(11 29 162)(12 30 163)(13 31 164)(14 32 165)(15 83 48)(16 84 49)(17 71 50)(18 72 51)(19 73 52)(20 74 53)(21 75 54)(22 76 55)(23 77 56)(24 78 43)(25 79 44)(26 80 45)(27 81 46)(28 82 47)(57 121 153)(58 122 154)(59 123 141)(60 124 142)(61 125 143)(62 126 144)(63 113 145)(64 114 146)(65 115 147)(66 116 148)(67 117 149)(68 118 150)(69 119 151)(70 120 152)(85 127 100)(86 128 101)(87 129 102)(88 130 103)(89 131 104)(90 132 105)(91 133 106)(92 134 107)(93 135 108)(94 136 109)(95 137 110)(96 138 111)(97 139 112)(98 140 99)
(1 145)(2 146)(3 147)(4 148)(5 149)(6 150)(7 151)(8 152)(9 153)(10 154)(11 141)(12 142)(13 143)(14 144)(15 106)(16 107)(17 108)(18 109)(19 110)(20 111)(21 112)(22 99)(23 100)(24 101)(25 102)(26 103)(27 104)(28 105)(29 123)(30 124)(31 125)(32 126)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 92)(50 93)(51 94)(52 95)(53 96)(54 97)(55 98)(56 85)(57 160)(58 161)(59 162)(60 163)(61 164)(62 165)(63 166)(64 167)(65 168)(66 155)(67 156)(68 157)(69 158)(70 159)(71 135)(72 136)(73 137)(74 138)(75 139)(76 140)(77 127)(78 128)(79 129)(80 130)(81 131)(82 132)(83 133)(84 134)
(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)
(1 96 8 89)(2 95 9 88)(3 94 10 87)(4 93 11 86)(5 92 12 85)(6 91 13 98)(7 90 14 97)(15 61 22 68)(16 60 23 67)(17 59 24 66)(18 58 25 65)(19 57 26 64)(20 70 27 63)(21 69 28 62)(29 128 36 135)(30 127 37 134)(31 140 38 133)(32 139 39 132)(33 138 40 131)(34 137 41 130)(35 136 42 129)(43 148 50 141)(44 147 51 154)(45 146 52 153)(46 145 53 152)(47 144 54 151)(48 143 55 150)(49 142 56 149)(71 123 78 116)(72 122 79 115)(73 121 80 114)(74 120 81 113)(75 119 82 126)(76 118 83 125)(77 117 84 124)(99 157 106 164)(100 156 107 163)(101 155 108 162)(102 168 109 161)(103 167 110 160)(104 166 111 159)(105 165 112 158)
G:=sub<Sym(168)| (1,152)(2,153)(3,154)(4,141)(5,142)(6,143)(7,144)(8,145)(9,146)(10,147)(11,148)(12,149)(13,150)(14,151)(15,140)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,133)(23,134)(24,135)(25,136)(26,137)(27,138)(28,139)(29,66)(30,67)(31,68)(32,69)(33,70)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(41,64)(42,65)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,99)(84,100)(113,159)(114,160)(115,161)(116,162)(117,163)(118,164)(119,165)(120,166)(121,167)(122,168)(123,155)(124,156)(125,157)(126,158), (1,33,166)(2,34,167)(3,35,168)(4,36,155)(5,37,156)(6,38,157)(7,39,158)(8,40,159)(9,41,160)(10,42,161)(11,29,162)(12,30,163)(13,31,164)(14,32,165)(15,83,48)(16,84,49)(17,71,50)(18,72,51)(19,73,52)(20,74,53)(21,75,54)(22,76,55)(23,77,56)(24,78,43)(25,79,44)(26,80,45)(27,81,46)(28,82,47)(57,121,153)(58,122,154)(59,123,141)(60,124,142)(61,125,143)(62,126,144)(63,113,145)(64,114,146)(65,115,147)(66,116,148)(67,117,149)(68,118,150)(69,119,151)(70,120,152)(85,127,100)(86,128,101)(87,129,102)(88,130,103)(89,131,104)(90,132,105)(91,133,106)(92,134,107)(93,135,108)(94,136,109)(95,137,110)(96,138,111)(97,139,112)(98,140,99), (1,145)(2,146)(3,147)(4,148)(5,149)(6,150)(7,151)(8,152)(9,153)(10,154)(11,141)(12,142)(13,143)(14,144)(15,106)(16,107)(17,108)(18,109)(19,110)(20,111)(21,112)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,123)(30,124)(31,125)(32,126)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,92)(50,93)(51,94)(52,95)(53,96)(54,97)(55,98)(56,85)(57,160)(58,161)(59,162)(60,163)(61,164)(62,165)(63,166)(64,167)(65,168)(66,155)(67,156)(68,157)(69,158)(70,159)(71,135)(72,136)(73,137)(74,138)(75,139)(76,140)(77,127)(78,128)(79,129)(80,130)(81,131)(82,132)(83,133)(84,134), (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), (1,96,8,89)(2,95,9,88)(3,94,10,87)(4,93,11,86)(5,92,12,85)(6,91,13,98)(7,90,14,97)(15,61,22,68)(16,60,23,67)(17,59,24,66)(18,58,25,65)(19,57,26,64)(20,70,27,63)(21,69,28,62)(29,128,36,135)(30,127,37,134)(31,140,38,133)(32,139,39,132)(33,138,40,131)(34,137,41,130)(35,136,42,129)(43,148,50,141)(44,147,51,154)(45,146,52,153)(46,145,53,152)(47,144,54,151)(48,143,55,150)(49,142,56,149)(71,123,78,116)(72,122,79,115)(73,121,80,114)(74,120,81,113)(75,119,82,126)(76,118,83,125)(77,117,84,124)(99,157,106,164)(100,156,107,163)(101,155,108,162)(102,168,109,161)(103,167,110,160)(104,166,111,159)(105,165,112,158)>;
G:=Group( (1,152)(2,153)(3,154)(4,141)(5,142)(6,143)(7,144)(8,145)(9,146)(10,147)(11,148)(12,149)(13,150)(14,151)(15,140)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,133)(23,134)(24,135)(25,136)(26,137)(27,138)(28,139)(29,66)(30,67)(31,68)(32,69)(33,70)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63)(41,64)(42,65)(43,93)(44,94)(45,95)(46,96)(47,97)(48,98)(49,85)(50,86)(51,87)(52,88)(53,89)(54,90)(55,91)(56,92)(71,101)(72,102)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,109)(80,110)(81,111)(82,112)(83,99)(84,100)(113,159)(114,160)(115,161)(116,162)(117,163)(118,164)(119,165)(120,166)(121,167)(122,168)(123,155)(124,156)(125,157)(126,158), (1,33,166)(2,34,167)(3,35,168)(4,36,155)(5,37,156)(6,38,157)(7,39,158)(8,40,159)(9,41,160)(10,42,161)(11,29,162)(12,30,163)(13,31,164)(14,32,165)(15,83,48)(16,84,49)(17,71,50)(18,72,51)(19,73,52)(20,74,53)(21,75,54)(22,76,55)(23,77,56)(24,78,43)(25,79,44)(26,80,45)(27,81,46)(28,82,47)(57,121,153)(58,122,154)(59,123,141)(60,124,142)(61,125,143)(62,126,144)(63,113,145)(64,114,146)(65,115,147)(66,116,148)(67,117,149)(68,118,150)(69,119,151)(70,120,152)(85,127,100)(86,128,101)(87,129,102)(88,130,103)(89,131,104)(90,132,105)(91,133,106)(92,134,107)(93,135,108)(94,136,109)(95,137,110)(96,138,111)(97,139,112)(98,140,99), (1,145)(2,146)(3,147)(4,148)(5,149)(6,150)(7,151)(8,152)(9,153)(10,154)(11,141)(12,142)(13,143)(14,144)(15,106)(16,107)(17,108)(18,109)(19,110)(20,111)(21,112)(22,99)(23,100)(24,101)(25,102)(26,103)(27,104)(28,105)(29,123)(30,124)(31,125)(32,126)(33,113)(34,114)(35,115)(36,116)(37,117)(38,118)(39,119)(40,120)(41,121)(42,122)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,92)(50,93)(51,94)(52,95)(53,96)(54,97)(55,98)(56,85)(57,160)(58,161)(59,162)(60,163)(61,164)(62,165)(63,166)(64,167)(65,168)(66,155)(67,156)(68,157)(69,158)(70,159)(71,135)(72,136)(73,137)(74,138)(75,139)(76,140)(77,127)(78,128)(79,129)(80,130)(81,131)(82,132)(83,133)(84,134), (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), (1,96,8,89)(2,95,9,88)(3,94,10,87)(4,93,11,86)(5,92,12,85)(6,91,13,98)(7,90,14,97)(15,61,22,68)(16,60,23,67)(17,59,24,66)(18,58,25,65)(19,57,26,64)(20,70,27,63)(21,69,28,62)(29,128,36,135)(30,127,37,134)(31,140,38,133)(32,139,39,132)(33,138,40,131)(34,137,41,130)(35,136,42,129)(43,148,50,141)(44,147,51,154)(45,146,52,153)(46,145,53,152)(47,144,54,151)(48,143,55,150)(49,142,56,149)(71,123,78,116)(72,122,79,115)(73,121,80,114)(74,120,81,113)(75,119,82,126)(76,118,83,125)(77,117,84,124)(99,157,106,164)(100,156,107,163)(101,155,108,162)(102,168,109,161)(103,167,110,160)(104,166,111,159)(105,165,112,158) );
G=PermutationGroup([[(1,152),(2,153),(3,154),(4,141),(5,142),(6,143),(7,144),(8,145),(9,146),(10,147),(11,148),(12,149),(13,150),(14,151),(15,140),(16,127),(17,128),(18,129),(19,130),(20,131),(21,132),(22,133),(23,134),(24,135),(25,136),(26,137),(27,138),(28,139),(29,66),(30,67),(31,68),(32,69),(33,70),(34,57),(35,58),(36,59),(37,60),(38,61),(39,62),(40,63),(41,64),(42,65),(43,93),(44,94),(45,95),(46,96),(47,97),(48,98),(49,85),(50,86),(51,87),(52,88),(53,89),(54,90),(55,91),(56,92),(71,101),(72,102),(73,103),(74,104),(75,105),(76,106),(77,107),(78,108),(79,109),(80,110),(81,111),(82,112),(83,99),(84,100),(113,159),(114,160),(115,161),(116,162),(117,163),(118,164),(119,165),(120,166),(121,167),(122,168),(123,155),(124,156),(125,157),(126,158)], [(1,33,166),(2,34,167),(3,35,168),(4,36,155),(5,37,156),(6,38,157),(7,39,158),(8,40,159),(9,41,160),(10,42,161),(11,29,162),(12,30,163),(13,31,164),(14,32,165),(15,83,48),(16,84,49),(17,71,50),(18,72,51),(19,73,52),(20,74,53),(21,75,54),(22,76,55),(23,77,56),(24,78,43),(25,79,44),(26,80,45),(27,81,46),(28,82,47),(57,121,153),(58,122,154),(59,123,141),(60,124,142),(61,125,143),(62,126,144),(63,113,145),(64,114,146),(65,115,147),(66,116,148),(67,117,149),(68,118,150),(69,119,151),(70,120,152),(85,127,100),(86,128,101),(87,129,102),(88,130,103),(89,131,104),(90,132,105),(91,133,106),(92,134,107),(93,135,108),(94,136,109),(95,137,110),(96,138,111),(97,139,112),(98,140,99)], [(1,145),(2,146),(3,147),(4,148),(5,149),(6,150),(7,151),(8,152),(9,153),(10,154),(11,141),(12,142),(13,143),(14,144),(15,106),(16,107),(17,108),(18,109),(19,110),(20,111),(21,112),(22,99),(23,100),(24,101),(25,102),(26,103),(27,104),(28,105),(29,123),(30,124),(31,125),(32,126),(33,113),(34,114),(35,115),(36,116),(37,117),(38,118),(39,119),(40,120),(41,121),(42,122),(43,86),(44,87),(45,88),(46,89),(47,90),(48,91),(49,92),(50,93),(51,94),(52,95),(53,96),(54,97),(55,98),(56,85),(57,160),(58,161),(59,162),(60,163),(61,164),(62,165),(63,166),(64,167),(65,168),(66,155),(67,156),(68,157),(69,158),(70,159),(71,135),(72,136),(73,137),(74,138),(75,139),(76,140),(77,127),(78,128),(79,129),(80,130),(81,131),(82,132),(83,133),(84,134)], [(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)], [(1,96,8,89),(2,95,9,88),(3,94,10,87),(4,93,11,86),(5,92,12,85),(6,91,13,98),(7,90,14,97),(15,61,22,68),(16,60,23,67),(17,59,24,66),(18,58,25,65),(19,57,26,64),(20,70,27,63),(21,69,28,62),(29,128,36,135),(30,127,37,134),(31,140,38,133),(32,139,39,132),(33,138,40,131),(34,137,41,130),(35,136,42,129),(43,148,50,141),(44,147,51,154),(45,146,52,153),(46,145,53,152),(47,144,54,151),(48,143,55,150),(49,142,56,149),(71,123,78,116),(72,122,79,115),(73,121,80,114),(74,120,81,113),(75,119,82,126),(76,118,83,125),(77,117,84,124),(99,157,106,164),(100,156,107,163),(101,155,108,162),(102,168,109,161),(103,167,110,160),(104,166,111,159),(105,165,112,158)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 7A | 7B | 7C | 12A | 12B | 12C | 12D | 14A | ··· | 14I | 14J | ··· | 14U | 21A | 21B | 21C | 42A | ··· | 42I |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 12 | 12 | 12 | 12 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | 21 | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 7 | 7 | 7 | 7 | 21 | 21 | 21 | 21 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | + | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | D7 | C4×S3 | Dic7 | D14 | D14 | S3×D7 | S3×Dic7 | C2×S3×D7 |
| kernel | C2×S3×Dic7 | S3×Dic7 | C6×Dic7 | C2×Dic21 | S3×C2×C14 | S3×C14 | C2×Dic7 | Dic7 | C2×C14 | C22×S3 | C14 | D6 | D6 | C2×C6 | C22 | C2 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 3 | 4 | 12 | 6 | 3 | 3 | 6 | 3 |
Matrix representation of C2×S3×Dic7 ►in GL4(𝔽337) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 336 | 0 |
| 0 | 0 | 0 | 336 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 335 | 28 |
| 0 | 0 | 36 | 1 |
| 336 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 |
| 0 | 0 | 336 | 0 |
| 0 | 0 | 36 | 1 |
| 304 | 336 | 0 | 0 |
| 78 | 227 | 0 | 0 |
| 0 | 0 | 336 | 0 |
| 0 | 0 | 0 | 336 |
| 334 | 41 | 0 | 0 |
| 271 | 3 | 0 | 0 |
| 0 | 0 | 189 | 0 |
| 0 | 0 | 0 | 189 |
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,336,0,0,0,0,336],[1,0,0,0,0,1,0,0,0,0,335,36,0,0,28,1],[336,0,0,0,0,336,0,0,0,0,336,36,0,0,0,1],[304,78,0,0,336,227,0,0,0,0,336,0,0,0,0,336],[334,271,0,0,41,3,0,0,0,0,189,0,0,0,0,189] >;
C2×S3×Dic7 in GAP, Magma, Sage, TeX
C_2\times S_3\times {\rm Dic}_7 % in TeX
G:=Group("C2xS3xDic7"); // GroupNames label
G:=SmallGroup(336,154);
// by ID
G=gap.SmallGroup(336,154);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,48,490,10373]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^14=1,e^2=d^7,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations