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## G = C2×S3×Dic7order 336 = 24·3·7

### Direct product of C2, S3 and Dic7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — C2×S3×Dic7
 Chief series C1 — C7 — C21 — C42 — C3×Dic7 — S3×Dic7 — C2×S3×Dic7
 Lower central C21 — C2×S3×Dic7
 Upper central C1 — C22

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

Smallest permutation representation of 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

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𝔽