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## G = S3×C22×C14order 336 = 24·3·7

### Direct product of C22×C14 and S3

Aliases: S3×C22×C14, C214C24, C424C23, C3⋊(C23×C14), C6⋊(C22×C14), (C22×C42)⋊7C2, (C22×C6)⋊5C14, (C2×C42)⋊14C22, (C2×C6)⋊4(C2×C14), SmallGroup(336,226)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C22×C14
 Chief series C1 — C3 — C21 — S3×C7 — S3×C14 — S3×C2×C14 — S3×C22×C14
 Lower central C3 — S3×C22×C14
 Upper central C1 — C22×C14

Generators and relations for S3×C22×C14
G = < a,b,c,d,e | a2=b2=c14=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 472 in 268 conjugacy classes, 166 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, S3, C6, C7, C23, C23, D6, C2×C6, C14, C14, C24, C21, C22×S3, C22×C6, C2×C14, C2×C14, S3×C7, C42, S3×C23, C22×C14, C22×C14, S3×C14, C2×C42, C23×C14, S3×C2×C14, C22×C42, S3×C22×C14
Quotients: C1, C2, C22, S3, C7, C23, D6, C14, C24, C22×S3, C2×C14, S3×C7, S3×C23, C22×C14, S3×C14, C23×C14, S3×C2×C14, S3×C22×C14

Smallest permutation representation of S3×C22×C14
On 168 points
Generators in S168
(1 139)(2 140)(3 127)(4 128)(5 129)(6 130)(7 131)(8 132)(9 133)(10 134)(11 135)(12 136)(13 137)(14 138)(15 56)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 99)(30 100)(31 101)(32 102)(33 103)(34 104)(35 105)(36 106)(37 107)(38 108)(39 109)(40 110)(41 111)(42 112)(57 167)(58 168)(59 155)(60 156)(61 157)(62 158)(63 159)(64 160)(65 161)(66 162)(67 163)(68 164)(69 165)(70 166)(71 142)(72 143)(73 144)(74 145)(75 146)(76 147)(77 148)(78 149)(79 150)(80 151)(81 152)(82 153)(83 154)(84 141)(85 124)(86 125)(87 126)(88 113)(89 114)(90 115)(91 116)(92 117)(93 118)(94 119)(95 120)(96 121)(97 122)(98 123)
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(29 119)(30 120)(31 121)(32 122)(33 123)(34 124)(35 125)(36 126)(37 113)(38 114)(39 115)(40 116)(41 117)(42 118)(43 133)(44 134)(45 135)(46 136)(47 137)(48 138)(49 139)(50 140)(51 127)(52 128)(53 129)(54 130)(55 131)(56 132)(57 149)(58 150)(59 151)(60 152)(61 153)(62 154)(63 141)(64 142)(65 143)(66 144)(67 145)(68 146)(69 147)(70 148)(71 160)(72 161)(73 162)(74 163)(75 164)(76 165)(77 166)(78 167)(79 168)(80 155)(81 156)(82 157)(83 158)(84 159)(85 104)(86 105)(87 106)(88 107)(89 108)(90 109)(91 110)(92 111)(93 112)(94 99)(95 100)(96 101)(97 102)(98 103)
(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 88 143)(2 89 144)(3 90 145)(4 91 146)(5 92 147)(6 93 148)(7 94 149)(8 95 150)(9 96 151)(10 97 152)(11 98 153)(12 85 154)(13 86 141)(14 87 142)(15 100 58)(16 101 59)(17 102 60)(18 103 61)(19 104 62)(20 105 63)(21 106 64)(22 107 65)(23 108 66)(24 109 67)(25 110 68)(26 111 69)(27 112 70)(28 99 57)(29 167 55)(30 168 56)(31 155 43)(32 156 44)(33 157 45)(34 158 46)(35 159 47)(36 160 48)(37 161 49)(38 162 50)(39 163 51)(40 164 52)(41 165 53)(42 166 54)(71 138 126)(72 139 113)(73 140 114)(74 127 115)(75 128 116)(76 129 117)(77 130 118)(78 131 119)(79 132 120)(80 133 121)(81 134 122)(82 135 123)(83 136 124)(84 137 125)
(1 139)(2 140)(3 127)(4 128)(5 129)(6 130)(7 131)(8 132)(9 133)(10 134)(11 135)(12 136)(13 137)(14 138)(15 56)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(71 87)(72 88)(73 89)(74 90)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)(81 97)(82 98)(83 85)(84 86)(99 167)(100 168)(101 155)(102 156)(103 157)(104 158)(105 159)(106 160)(107 161)(108 162)(109 163)(110 164)(111 165)(112 166)(113 143)(114 144)(115 145)(116 146)(117 147)(118 148)(119 149)(120 150)(121 151)(122 152)(123 153)(124 154)(125 141)(126 142)

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

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

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

168 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3 6A ··· 6G 7A ··· 7F 14A ··· 14AP 14AQ ··· 14CL 21A ··· 21F 42A ··· 42AP order 1 2 ··· 2 2 ··· 2 3 6 ··· 6 7 ··· 7 14 ··· 14 14 ··· 14 21 ··· 21 42 ··· 42 size 1 1 ··· 1 3 ··· 3 2 2 ··· 2 1 ··· 1 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2

168 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C7 C14 C14 S3 D6 S3×C7 S3×C14 kernel S3×C22×C14 S3×C2×C14 C22×C42 S3×C23 C22×S3 C22×C6 C22×C14 C2×C14 C23 C22 # reps 1 14 1 6 84 6 1 7 6 42

Matrix representation of S3×C22×C14 in GL4(𝔽43) generated by

 1 0 0 0 0 42 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 42 0 0 0 0 42 0 0 0 0 42
,
 42 0 0 0 0 1 0 0 0 0 35 0 0 0 0 35
,
 1 0 0 0 0 1 0 0 0 0 42 42 0 0 1 0
,
 42 0 0 0 0 1 0 0 0 0 1 0 0 0 42 42
G:=sub<GL(4,GF(43))| [1,0,0,0,0,42,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,42,0,0,0,0,42,0,0,0,0,42],[42,0,0,0,0,1,0,0,0,0,35,0,0,0,0,35],[1,0,0,0,0,1,0,0,0,0,42,1,0,0,42,0],[42,0,0,0,0,1,0,0,0,0,1,42,0,0,0,42] >;

S3×C22×C14 in GAP, Magma, Sage, TeX

S_3\times C_2^2\times C_{14}
% in TeX

G:=Group("S3xC2^2xC14");
// GroupNames label

G:=SmallGroup(336,226);
// by ID

G=gap.SmallGroup(336,226);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-7,-3,8069]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^14=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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