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## G = C2×Dic3×D7order 336 = 24·3·7

### Direct product of C2, Dic3 and D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C2×Dic3×D7
 Chief series C1 — C7 — C21 — C42 — C6×D7 — Dic3×D7 — C2×Dic3×D7
 Lower central C21 — C2×Dic3×D7
 Upper central C1 — C22

Generators and relations for C2×Dic3×D7
G = < a,b,c,d,e | a2=b6=d7=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 476 in 108 conjugacy classes, 56 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C7, C2×C4, C23, Dic3, Dic3, C2×C6, C2×C6, D7, C14, C14, C22×C4, C21, C2×Dic3, C2×Dic3, C22×C6, Dic7, C28, D14, C2×C14, C3×D7, C42, C42, C22×Dic3, C4×D7, C2×Dic7, C2×C28, C22×D7, C7×Dic3, Dic21, C6×D7, C2×C42, C2×C4×D7, Dic3×D7, Dic3×C14, C2×Dic21, C2×C6×D7, C2×Dic3×D7
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, D7, C22×C4, C2×Dic3, C22×S3, D14, C22×Dic3, C4×D7, C22×D7, S3×D7, C2×C4×D7, Dic3×D7, C2×S3×D7, C2×Dic3×D7

Smallest permutation representation of C2×Dic3×D7
On 168 points
Generators in S168
(1 48)(2 49)(3 43)(4 44)(5 45)(6 46)(7 47)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 81)(40 82)(41 83)(42 84)(85 127)(86 128)(87 129)(88 130)(89 131)(90 132)(91 133)(92 134)(93 135)(94 136)(95 137)(96 138)(97 139)(98 140)(99 141)(100 142)(101 143)(102 144)(103 145)(104 146)(105 147)(106 148)(107 149)(108 150)(109 151)(110 152)(111 153)(112 154)(113 155)(114 156)(115 157)(116 158)(117 159)(118 160)(119 161)(120 162)(121 163)(122 164)(123 165)(124 166)(125 167)(126 168)
(1 41 13 27 20 34)(2 42 14 28 21 35)(3 36 8 22 15 29)(4 37 9 23 16 30)(5 38 10 24 17 31)(6 39 11 25 18 32)(7 40 12 26 19 33)(43 78 50 64 57 71)(44 79 51 65 58 72)(45 80 52 66 59 73)(46 81 53 67 60 74)(47 82 54 68 61 75)(48 83 55 69 62 76)(49 84 56 70 63 77)(85 113 99 106 92 120)(86 114 100 107 93 121)(87 115 101 108 94 122)(88 116 102 109 95 123)(89 117 103 110 96 124)(90 118 104 111 97 125)(91 119 105 112 98 126)(127 155 141 148 134 162)(128 156 142 149 135 163)(129 157 143 150 136 164)(130 158 144 151 137 165)(131 159 145 152 138 166)(132 160 146 153 139 167)(133 161 147 154 140 168)
(1 153 27 132)(2 154 28 133)(3 148 22 127)(4 149 23 128)(5 150 24 129)(6 151 25 130)(7 152 26 131)(8 155 29 134)(9 156 30 135)(10 157 31 136)(11 158 32 137)(12 159 33 138)(13 160 34 139)(14 161 35 140)(15 162 36 141)(16 163 37 142)(17 164 38 143)(18 165 39 144)(19 166 40 145)(20 167 41 146)(21 168 42 147)(43 106 64 85)(44 107 65 86)(45 108 66 87)(46 109 67 88)(47 110 68 89)(48 111 69 90)(49 112 70 91)(50 113 71 92)(51 114 72 93)(52 115 73 94)(53 116 74 95)(54 117 75 96)(55 118 76 97)(56 119 77 98)(57 120 78 99)(58 121 79 100)(59 122 80 101)(60 123 81 102)(61 124 82 103)(62 125 83 104)(63 126 84 105)
(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 68)(2 67)(3 66)(4 65)(5 64)(6 70)(7 69)(8 73)(9 72)(10 71)(11 77)(12 76)(13 75)(14 74)(15 80)(16 79)(17 78)(18 84)(19 83)(20 82)(21 81)(22 45)(23 44)(24 43)(25 49)(26 48)(27 47)(28 46)(29 52)(30 51)(31 50)(32 56)(33 55)(34 54)(35 53)(36 59)(37 58)(38 57)(39 63)(40 62)(41 61)(42 60)(85 150)(86 149)(87 148)(88 154)(89 153)(90 152)(91 151)(92 157)(93 156)(94 155)(95 161)(96 160)(97 159)(98 158)(99 164)(100 163)(101 162)(102 168)(103 167)(104 166)(105 165)(106 129)(107 128)(108 127)(109 133)(110 132)(111 131)(112 130)(113 136)(114 135)(115 134)(116 140)(117 139)(118 138)(119 137)(120 143)(121 142)(122 141)(123 147)(124 146)(125 145)(126 144)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F 6G 7A 7B 7C 14A ··· 14I 21A 21B 21C 28A ··· 28L 42A ··· 42I order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 7 7 7 14 ··· 14 21 21 21 28 ··· 28 42 ··· 42 size 1 1 1 1 7 7 7 7 2 3 3 3 3 21 21 21 21 2 2 2 14 14 14 14 2 2 2 2 ··· 2 4 4 4 6 ··· 6 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 Dic3 D6 D6 D7 D14 D14 C4×D7 S3×D7 Dic3×D7 C2×S3×D7 kernel C2×Dic3×D7 Dic3×D7 Dic3×C14 C2×Dic21 C2×C6×D7 C6×D7 C22×D7 D14 D14 C2×C14 C2×Dic3 Dic3 C2×C6 C6 C22 C2 C2 # reps 1 4 1 1 1 8 1 4 2 1 3 6 3 12 3 6 3

Matrix representation of C2×Dic3×D7 in GL4(𝔽337) generated by

 1 0 0 0 0 1 0 0 0 0 336 0 0 0 0 336
,
 336 0 0 0 0 336 0 0 0 0 336 1 0 0 336 0
,
 148 0 0 0 0 148 0 0 0 0 54 271 0 0 325 283
,
 109 1 0 0 251 194 0 0 0 0 1 0 0 0 0 1
,
 143 34 0 0 251 194 0 0 0 0 336 0 0 0 0 336
G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,336,0,0,0,0,336],[336,0,0,0,0,336,0,0,0,0,336,336,0,0,1,0],[148,0,0,0,0,148,0,0,0,0,54,325,0,0,271,283],[109,251,0,0,1,194,0,0,0,0,1,0,0,0,0,1],[143,251,0,0,34,194,0,0,0,0,336,0,0,0,0,336] >;

C2×Dic3×D7 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_3\times D_7
% in TeX

G:=Group("C2xDic3xD7");
// GroupNames label

G:=SmallGroup(336,151);
// by ID

G=gap.SmallGroup(336,151);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-7,55,490,10373]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^6=d^7=e^2=1,c^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽