direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×C22×C6, C21⋊3C24, C42⋊3C23, C7⋊3(C23×C6), C14⋊3(C22×C6), (C22×C42)⋊5C2, (C22×C14)⋊11C6, (C2×C42)⋊12C22, (C2×C14)⋊13(C2×C6), SmallGroup(336,225)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7×C22×C6 |
Generators and relations for D7×C22×C6
G = < a,b,c,d,e | a2=b2=c6=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 880 in 268 conjugacy classes, 166 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, C6, C6, C7, C23, C23, C2×C6, C2×C6, D7, C14, C24, C21, C22×C6, C22×C6, D14, C2×C14, C3×D7, C42, C23×C6, C22×D7, C22×C14, C6×D7, C2×C42, C23×D7, C2×C6×D7, C22×C42, D7×C22×C6
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, D7, C24, C22×C6, D14, C3×D7, C23×C6, C22×D7, C6×D7, C23×D7, C2×C6×D7, D7×C22×C6
►On 168 points
Generators in S168
(1 90)(2 91)(3 85)(4 86)(5 87)(6 88)(7 89)(8 92)(9 93)(10 94)(11 95)(12 96)(13 97)(14 98)(15 99)(16 100)(17 101)(18 102)(19 103)(20 104)(21 105)(22 106)(23 107)(24 108)(25 109)(26 110)(27 111)(28 112)(29 113)(30 114)(31 115)(32 116)(33 117)(34 118)(35 119)(36 120)(37 121)(38 122)(39 123)(40 124)(41 125)(42 126)(43 127)(44 128)(45 129)(46 130)(47 131)(48 132)(49 133)(50 134)(51 135)(52 136)(53 137)(54 138)(55 139)(56 140)(57 141)(58 142)(59 143)(60 144)(61 145)(62 146)(63 147)(64 148)(65 149)(66 150)(67 151)(68 152)(69 153)(70 154)(71 155)(72 156)(73 157)(74 158)(75 159)(76 160)(77 161)(78 162)(79 163)(80 164)(81 165)(82 166)(83 167)(84 168)
(1 69)(2 70)(3 64)(4 65)(5 66)(6 67)(7 68)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 57)(37 58)(38 59)(39 60)(40 61)(41 62)(42 63)(85 148)(86 149)(87 150)(88 151)(89 152)(90 153)(91 154)(92 155)(93 156)(94 157)(95 158)(96 159)(97 160)(98 161)(99 162)(100 163)(101 164)(102 165)(103 166)(104 167)(105 168)(106 127)(107 128)(108 129)(109 130)(110 131)(111 132)(112 133)(113 134)(114 135)(115 136)(116 137)(117 138)(118 139)(119 140)(120 141)(121 142)(122 143)(123 144)(124 145)(125 146)(126 147)
(1 34 20 27 13 41)(2 35 21 28 14 42)(3 29 15 22 8 36)(4 30 16 23 9 37)(5 31 17 24 10 38)(6 32 18 25 11 39)(7 33 19 26 12 40)(43 71 57 64 50 78)(44 72 58 65 51 79)(45 73 59 66 52 80)(46 74 60 67 53 81)(47 75 61 68 54 82)(48 76 62 69 55 83)(49 77 63 70 56 84)(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 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 47)(2 46)(3 45)(4 44)(5 43)(6 49)(7 48)(8 52)(9 51)(10 50)(11 56)(12 55)(13 54)(14 53)(15 59)(16 58)(17 57)(18 63)(19 62)(20 61)(21 60)(22 66)(23 65)(24 64)(25 70)(26 69)(27 68)(28 67)(29 73)(30 72)(31 71)(32 77)(33 76)(34 75)(35 74)(36 80)(37 79)(38 78)(39 84)(40 83)(41 82)(42 81)(85 129)(86 128)(87 127)(88 133)(89 132)(90 131)(91 130)(92 136)(93 135)(94 134)(95 140)(96 139)(97 138)(98 137)(99 143)(100 142)(101 141)(102 147)(103 146)(104 145)(105 144)(106 150)(107 149)(108 148)(109 154)(110 153)(111 152)(112 151)(113 157)(114 156)(115 155)(116 161)(117 160)(118 159)(119 158)(120 164)(121 163)(122 162)(123 168)(124 167)(125 166)(126 165)
G:=sub<Sym(168)| (1,90)(2,91)(3,85)(4,86)(5,87)(6,88)(7,89)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,113)(30,114)(31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,121)(38,122)(39,123)(40,124)(41,125)(42,126)(43,127)(44,128)(45,129)(46,130)(47,131)(48,132)(49,133)(50,134)(51,135)(52,136)(53,137)(54,138)(55,139)(56,140)(57,141)(58,142)(59,143)(60,144)(61,145)(62,146)(63,147)(64,148)(65,149)(66,150)(67,151)(68,152)(69,153)(70,154)(71,155)(72,156)(73,157)(74,158)(75,159)(76,160)(77,161)(78,162)(79,163)(80,164)(81,165)(82,166)(83,167)(84,168), (1,69)(2,70)(3,64)(4,65)(5,66)(6,67)(7,68)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(85,148)(86,149)(87,150)(88,151)(89,152)(90,153)(91,154)(92,155)(93,156)(94,157)(95,158)(96,159)(97,160)(98,161)(99,162)(100,163)(101,164)(102,165)(103,166)(104,167)(105,168)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,141)(121,142)(122,143)(123,144)(124,145)(125,146)(126,147), (1,34,20,27,13,41)(2,35,21,28,14,42)(3,29,15,22,8,36)(4,30,16,23,9,37)(5,31,17,24,10,38)(6,32,18,25,11,39)(7,33,19,26,12,40)(43,71,57,64,50,78)(44,72,58,65,51,79)(45,73,59,66,52,80)(46,74,60,67,53,81)(47,75,61,68,54,82)(48,76,62,69,55,83)(49,77,63,70,56,84)(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), (1,2,3,4,5,6,7)(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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,49)(7,48)(8,52)(9,51)(10,50)(11,56)(12,55)(13,54)(14,53)(15,59)(16,58)(17,57)(18,63)(19,62)(20,61)(21,60)(22,66)(23,65)(24,64)(25,70)(26,69)(27,68)(28,67)(29,73)(30,72)(31,71)(32,77)(33,76)(34,75)(35,74)(36,80)(37,79)(38,78)(39,84)(40,83)(41,82)(42,81)(85,129)(86,128)(87,127)(88,133)(89,132)(90,131)(91,130)(92,136)(93,135)(94,134)(95,140)(96,139)(97,138)(98,137)(99,143)(100,142)(101,141)(102,147)(103,146)(104,145)(105,144)(106,150)(107,149)(108,148)(109,154)(110,153)(111,152)(112,151)(113,157)(114,156)(115,155)(116,161)(117,160)(118,159)(119,158)(120,164)(121,163)(122,162)(123,168)(124,167)(125,166)(126,165)>;
G:=Group( (1,90)(2,91)(3,85)(4,86)(5,87)(6,88)(7,89)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,99)(16,100)(17,101)(18,102)(19,103)(20,104)(21,105)(22,106)(23,107)(24,108)(25,109)(26,110)(27,111)(28,112)(29,113)(30,114)(31,115)(32,116)(33,117)(34,118)(35,119)(36,120)(37,121)(38,122)(39,123)(40,124)(41,125)(42,126)(43,127)(44,128)(45,129)(46,130)(47,131)(48,132)(49,133)(50,134)(51,135)(52,136)(53,137)(54,138)(55,139)(56,140)(57,141)(58,142)(59,143)(60,144)(61,145)(62,146)(63,147)(64,148)(65,149)(66,150)(67,151)(68,152)(69,153)(70,154)(71,155)(72,156)(73,157)(74,158)(75,159)(76,160)(77,161)(78,162)(79,163)(80,164)(81,165)(82,166)(83,167)(84,168), (1,69)(2,70)(3,64)(4,65)(5,66)(6,67)(7,68)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(85,148)(86,149)(87,150)(88,151)(89,152)(90,153)(91,154)(92,155)(93,156)(94,157)(95,158)(96,159)(97,160)(98,161)(99,162)(100,163)(101,164)(102,165)(103,166)(104,167)(105,168)(106,127)(107,128)(108,129)(109,130)(110,131)(111,132)(112,133)(113,134)(114,135)(115,136)(116,137)(117,138)(118,139)(119,140)(120,141)(121,142)(122,143)(123,144)(124,145)(125,146)(126,147), (1,34,20,27,13,41)(2,35,21,28,14,42)(3,29,15,22,8,36)(4,30,16,23,9,37)(5,31,17,24,10,38)(6,32,18,25,11,39)(7,33,19,26,12,40)(43,71,57,64,50,78)(44,72,58,65,51,79)(45,73,59,66,52,80)(46,74,60,67,53,81)(47,75,61,68,54,82)(48,76,62,69,55,83)(49,77,63,70,56,84)(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), (1,2,3,4,5,6,7)(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), (1,47)(2,46)(3,45)(4,44)(5,43)(6,49)(7,48)(8,52)(9,51)(10,50)(11,56)(12,55)(13,54)(14,53)(15,59)(16,58)(17,57)(18,63)(19,62)(20,61)(21,60)(22,66)(23,65)(24,64)(25,70)(26,69)(27,68)(28,67)(29,73)(30,72)(31,71)(32,77)(33,76)(34,75)(35,74)(36,80)(37,79)(38,78)(39,84)(40,83)(41,82)(42,81)(85,129)(86,128)(87,127)(88,133)(89,132)(90,131)(91,130)(92,136)(93,135)(94,134)(95,140)(96,139)(97,138)(98,137)(99,143)(100,142)(101,141)(102,147)(103,146)(104,145)(105,144)(106,150)(107,149)(108,148)(109,154)(110,153)(111,152)(112,151)(113,157)(114,156)(115,155)(116,161)(117,160)(118,159)(119,158)(120,164)(121,163)(122,162)(123,168)(124,167)(125,166)(126,165) );
G=PermutationGroup([[(1,90),(2,91),(3,85),(4,86),(5,87),(6,88),(7,89),(8,92),(9,93),(10,94),(11,95),(12,96),(13,97),(14,98),(15,99),(16,100),(17,101),(18,102),(19,103),(20,104),(21,105),(22,106),(23,107),(24,108),(25,109),(26,110),(27,111),(28,112),(29,113),(30,114),(31,115),(32,116),(33,117),(34,118),(35,119),(36,120),(37,121),(38,122),(39,123),(40,124),(41,125),(42,126),(43,127),(44,128),(45,129),(46,130),(47,131),(48,132),(49,133),(50,134),(51,135),(52,136),(53,137),(54,138),(55,139),(56,140),(57,141),(58,142),(59,143),(60,144),(61,145),(62,146),(63,147),(64,148),(65,149),(66,150),(67,151),(68,152),(69,153),(70,154),(71,155),(72,156),(73,157),(74,158),(75,159),(76,160),(77,161),(78,162),(79,163),(80,164),(81,165),(82,166),(83,167),(84,168)], [(1,69),(2,70),(3,64),(4,65),(5,66),(6,67),(7,68),(8,71),(9,72),(10,73),(11,74),(12,75),(13,76),(14,77),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,57),(37,58),(38,59),(39,60),(40,61),(41,62),(42,63),(85,148),(86,149),(87,150),(88,151),(89,152),(90,153),(91,154),(92,155),(93,156),(94,157),(95,158),(96,159),(97,160),(98,161),(99,162),(100,163),(101,164),(102,165),(103,166),(104,167),(105,168),(106,127),(107,128),(108,129),(109,130),(110,131),(111,132),(112,133),(113,134),(114,135),(115,136),(116,137),(117,138),(118,139),(119,140),(120,141),(121,142),(122,143),(123,144),(124,145),(125,146),(126,147)], [(1,34,20,27,13,41),(2,35,21,28,14,42),(3,29,15,22,8,36),(4,30,16,23,9,37),(5,31,17,24,10,38),(6,32,18,25,11,39),(7,33,19,26,12,40),(43,71,57,64,50,78),(44,72,58,65,51,79),(45,73,59,66,52,80),(46,74,60,67,53,81),(47,75,61,68,54,82),(48,76,62,69,55,83),(49,77,63,70,56,84),(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)], [(1,2,3,4,5,6,7),(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)], [(1,47),(2,46),(3,45),(4,44),(5,43),(6,49),(7,48),(8,52),(9,51),(10,50),(11,56),(12,55),(13,54),(14,53),(15,59),(16,58),(17,57),(18,63),(19,62),(20,61),(21,60),(22,66),(23,65),(24,64),(25,70),(26,69),(27,68),(28,67),(29,73),(30,72),(31,71),(32,77),(33,76),(34,75),(35,74),(36,80),(37,79),(38,78),(39,84),(40,83),(41,82),(42,81),(85,129),(86,128),(87,127),(88,133),(89,132),(90,131),(91,130),(92,136),(93,135),(94,134),(95,140),(96,139),(97,138),(98,137),(99,143),(100,142),(101,141),(102,147),(103,146),(104,145),(105,144),(106,150),(107,149),(108,148),(109,154),(110,153),(111,152),(112,151),(113,157),(114,156),(115,155),(116,161),(117,160),(118,159),(119,158),(120,164),(121,163),(122,162),(123,168),(124,167),(125,166),(126,165)]])
120 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3A | 3B | 6A | ··· | 6N | 6O | ··· | 6AD | 7A | 7B | 7C | 14A | ··· | 14U | 21A | ··· | 21F | 42A | ··· | 42AP |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 3 | 6 | ··· | 6 | 6 | ··· | 6 | 7 | 7 | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | ··· | 1 | 7 | ··· | 7 | 1 | 1 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D7 | D14 | C3×D7 | C6×D7 |
| kernel | D7×C22×C6 | C2×C6×D7 | C22×C42 | C23×D7 | C22×D7 | C22×C14 | C22×C6 | C2×C6 | C23 | C22 |
| # reps | 1 | 14 | 1 | 2 | 28 | 2 | 3 | 21 | 6 | 42 |
Matrix representation of D7×C22×C6 ►in GL4(𝔽43) generated by
| 42 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 |
| 0 | 0 | 42 | 0 |
| 0 | 0 | 0 | 42 |
| 1 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 |
| 0 | 0 | 42 | 0 |
| 0 | 0 | 0 | 42 |
| 42 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 42 | 15 |
| 1 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 |
| 0 | 0 | 34 | 15 |
| 0 | 0 | 9 | 9 |
G:=sub<GL(4,GF(43))| [42,0,0,0,0,42,0,0,0,0,42,0,0,0,0,42],[1,0,0,0,0,42,0,0,0,0,42,0,0,0,0,42],[42,0,0,0,0,42,0,0,0,0,6,0,0,0,0,6],[1,0,0,0,0,1,0,0,0,0,0,42,0,0,1,15],[1,0,0,0,0,42,0,0,0,0,34,9,0,0,15,9] >;
D7×C22×C6 in GAP, Magma, Sage, TeX
D_7\times C_2^2\times C_6
% in TeX
G:=Group("D7xC2^2xC6"); // GroupNames label
G:=SmallGroup(336,225);
// by ID
G=gap.SmallGroup(336,225);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,10373]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^6=d^7=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