direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D4⋊D7, C21⋊8D8, D28⋊8C6, C42.40D4, C12.36D14, C84.36C22, C7⋊C8⋊8C6, D4⋊(C3×D7), C7⋊5(C3×D8), (C7×D4)⋊7C6, (C3×D4)⋊4D7, C4.1(C6×D7), (C3×D28)⋊8C2, (D4×C21)⋊4C2, C28.22(C2×C6), C14.23(C3×D4), C6.23(C7⋊D4), (C3×C7⋊C8)⋊8C2, C2.4(C3×C7⋊D4), SmallGroup(336,69)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D4⋊D7
G = < a,b,c,d,e | a3=b4=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=d-1 >
Smallest permutation representation of C3×D4⋊D7
►On 168 points
Generators in S168
(1 57 29)(2 58 30)(3 59 31)(4 60 32)(5 61 33)(6 62 34)(7 63 35)(8 64 36)(9 65 37)(10 66 38)(11 67 39)(12 68 40)(13 69 41)(14 70 42)(15 71 43)(16 72 44)(17 73 45)(18 74 46)(19 75 47)(20 76 48)(21 77 49)(22 78 50)(23 79 51)(24 80 52)(25 81 53)(26 82 54)(27 83 55)(28 84 56)(85 141 113)(86 142 114)(87 143 115)(88 144 116)(89 145 117)(90 146 118)(91 147 119)(92 148 120)(93 149 121)(94 150 122)(95 151 123)(96 152 124)(97 153 125)(98 154 126)(99 155 127)(100 156 128)(101 157 129)(102 158 130)(103 159 131)(104 160 132)(105 161 133)(106 162 134)(107 163 135)(108 164 136)(109 165 137)(110 166 138)(111 167 139)(112 168 140)
(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)(29 50 36 43)(30 51 37 44)(31 52 38 45)(32 53 39 46)(33 54 40 47)(34 55 41 48)(35 56 42 49)(57 78 64 71)(58 79 65 72)(59 80 66 73)(60 81 67 74)(61 82 68 75)(62 83 69 76)(63 84 70 77)(85 99 92 106)(86 100 93 107)(87 101 94 108)(88 102 95 109)(89 103 96 110)(90 104 97 111)(91 105 98 112)(113 127 120 134)(114 128 121 135)(115 129 122 136)(116 130 123 137)(117 131 124 138)(118 132 125 139)(119 133 126 140)(141 155 148 162)(142 156 149 163)(143 157 150 164)(144 158 151 165)(145 159 152 166)(146 160 153 167)(147 161 154 168)
(1 106)(2 107)(3 108)(4 109)(5 110)(6 111)(7 112)(8 99)(9 100)(10 101)(11 102)(12 103)(13 104)(14 105)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 98)(29 134)(30 135)(31 136)(32 137)(33 138)(34 139)(35 140)(36 127)(37 128)(38 129)(39 130)(40 131)(41 132)(42 133)(43 113)(44 114)(45 115)(46 116)(47 117)(48 118)(49 119)(50 120)(51 121)(52 122)(53 123)(54 124)(55 125)(56 126)(57 162)(58 163)(59 164)(60 165)(61 166)(62 167)(63 168)(64 155)(65 156)(66 157)(67 158)(68 159)(69 160)(70 161)(71 141)(72 142)(73 143)(74 144)(75 145)(76 146)(77 147)(78 148)(79 149)(80 150)(81 151)(82 152)(83 153)(84 154)
(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 7)(2 6)(3 5)(8 14)(9 13)(10 12)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)(29 35)(30 34)(31 33)(36 42)(37 41)(38 40)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(57 63)(58 62)(59 61)(64 70)(65 69)(66 68)(71 84)(72 83)(73 82)(74 81)(75 80)(76 79)(77 78)(85 105)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)(92 112)(93 111)(94 110)(95 109)(96 108)(97 107)(98 106)(113 133)(114 132)(115 131)(116 130)(117 129)(118 128)(119 127)(120 140)(121 139)(122 138)(123 137)(124 136)(125 135)(126 134)(141 161)(142 160)(143 159)(144 158)(145 157)(146 156)(147 155)(148 168)(149 167)(150 166)(151 165)(152 164)(153 163)(154 162)
G:=sub<Sym(168)| (1,57,29)(2,58,30)(3,59,31)(4,60,32)(5,61,33)(6,62,34)(7,63,35)(8,64,36)(9,65,37)(10,66,38)(11,67,39)(12,68,40)(13,69,41)(14,70,42)(15,71,43)(16,72,44)(17,73,45)(18,74,46)(19,75,47)(20,76,48)(21,77,49)(22,78,50)(23,79,51)(24,80,52)(25,81,53)(26,82,54)(27,83,55)(28,84,56)(85,141,113)(86,142,114)(87,143,115)(88,144,116)(89,145,117)(90,146,118)(91,147,119)(92,148,120)(93,149,121)(94,150,122)(95,151,123)(96,152,124)(97,153,125)(98,154,126)(99,155,127)(100,156,128)(101,157,129)(102,158,130)(103,159,131)(104,160,132)(105,161,133)(106,162,134)(107,163,135)(108,164,136)(109,165,137)(110,166,138)(111,167,139)(112,168,140), (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21)(29,50,36,43)(30,51,37,44)(31,52,38,45)(32,53,39,46)(33,54,40,47)(34,55,41,48)(35,56,42,49)(57,78,64,71)(58,79,65,72)(59,80,66,73)(60,81,67,74)(61,82,68,75)(62,83,69,76)(63,84,70,77)(85,99,92,106)(86,100,93,107)(87,101,94,108)(88,102,95,109)(89,103,96,110)(90,104,97,111)(91,105,98,112)(113,127,120,134)(114,128,121,135)(115,129,122,136)(116,130,123,137)(117,131,124,138)(118,132,125,139)(119,133,126,140)(141,155,148,162)(142,156,149,163)(143,157,150,164)(144,158,151,165)(145,159,152,166)(146,160,153,167)(147,161,154,168), (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,99)(9,100)(10,101)(11,102)(12,103)(13,104)(14,105)(15,85)(16,86)(17,87)(18,88)(19,89)(20,90)(21,91)(22,92)(23,93)(24,94)(25,95)(26,96)(27,97)(28,98)(29,134)(30,135)(31,136)(32,137)(33,138)(34,139)(35,140)(36,127)(37,128)(38,129)(39,130)(40,131)(41,132)(42,133)(43,113)(44,114)(45,115)(46,116)(47,117)(48,118)(49,119)(50,120)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(57,162)(58,163)(59,164)(60,165)(61,166)(62,167)(63,168)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,161)(71,141)(72,142)(73,143)(74,144)(75,145)(76,146)(77,147)(78,148)(79,149)(80,150)(81,151)(82,152)(83,153)(84,154), (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), (1,7)(2,6)(3,5)(8,14)(9,13)(10,12)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(29,35)(30,34)(31,33)(36,42)(37,41)(38,40)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(57,63)(58,62)(59,61)(64,70)(65,69)(66,68)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)(85,105)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,112)(93,111)(94,110)(95,109)(96,108)(97,107)(98,106)(113,133)(114,132)(115,131)(116,130)(117,129)(118,128)(119,127)(120,140)(121,139)(122,138)(123,137)(124,136)(125,135)(126,134)(141,161)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,168)(149,167)(150,166)(151,165)(152,164)(153,163)(154,162)>;
G:=Group( (1,57,29)(2,58,30)(3,59,31)(4,60,32)(5,61,33)(6,62,34)(7,63,35)(8,64,36)(9,65,37)(10,66,38)(11,67,39)(12,68,40)(13,69,41)(14,70,42)(15,71,43)(16,72,44)(17,73,45)(18,74,46)(19,75,47)(20,76,48)(21,77,49)(22,78,50)(23,79,51)(24,80,52)(25,81,53)(26,82,54)(27,83,55)(28,84,56)(85,141,113)(86,142,114)(87,143,115)(88,144,116)(89,145,117)(90,146,118)(91,147,119)(92,148,120)(93,149,121)(94,150,122)(95,151,123)(96,152,124)(97,153,125)(98,154,126)(99,155,127)(100,156,128)(101,157,129)(102,158,130)(103,159,131)(104,160,132)(105,161,133)(106,162,134)(107,163,135)(108,164,136)(109,165,137)(110,166,138)(111,167,139)(112,168,140), (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21)(29,50,36,43)(30,51,37,44)(31,52,38,45)(32,53,39,46)(33,54,40,47)(34,55,41,48)(35,56,42,49)(57,78,64,71)(58,79,65,72)(59,80,66,73)(60,81,67,74)(61,82,68,75)(62,83,69,76)(63,84,70,77)(85,99,92,106)(86,100,93,107)(87,101,94,108)(88,102,95,109)(89,103,96,110)(90,104,97,111)(91,105,98,112)(113,127,120,134)(114,128,121,135)(115,129,122,136)(116,130,123,137)(117,131,124,138)(118,132,125,139)(119,133,126,140)(141,155,148,162)(142,156,149,163)(143,157,150,164)(144,158,151,165)(145,159,152,166)(146,160,153,167)(147,161,154,168), (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,99)(9,100)(10,101)(11,102)(12,103)(13,104)(14,105)(15,85)(16,86)(17,87)(18,88)(19,89)(20,90)(21,91)(22,92)(23,93)(24,94)(25,95)(26,96)(27,97)(28,98)(29,134)(30,135)(31,136)(32,137)(33,138)(34,139)(35,140)(36,127)(37,128)(38,129)(39,130)(40,131)(41,132)(42,133)(43,113)(44,114)(45,115)(46,116)(47,117)(48,118)(49,119)(50,120)(51,121)(52,122)(53,123)(54,124)(55,125)(56,126)(57,162)(58,163)(59,164)(60,165)(61,166)(62,167)(63,168)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,161)(71,141)(72,142)(73,143)(74,144)(75,145)(76,146)(77,147)(78,148)(79,149)(80,150)(81,151)(82,152)(83,153)(84,154), (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), (1,7)(2,6)(3,5)(8,14)(9,13)(10,12)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(29,35)(30,34)(31,33)(36,42)(37,41)(38,40)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(57,63)(58,62)(59,61)(64,70)(65,69)(66,68)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)(85,105)(86,104)(87,103)(88,102)(89,101)(90,100)(91,99)(92,112)(93,111)(94,110)(95,109)(96,108)(97,107)(98,106)(113,133)(114,132)(115,131)(116,130)(117,129)(118,128)(119,127)(120,140)(121,139)(122,138)(123,137)(124,136)(125,135)(126,134)(141,161)(142,160)(143,159)(144,158)(145,157)(146,156)(147,155)(148,168)(149,167)(150,166)(151,165)(152,164)(153,163)(154,162) );
G=PermutationGroup([[(1,57,29),(2,58,30),(3,59,31),(4,60,32),(5,61,33),(6,62,34),(7,63,35),(8,64,36),(9,65,37),(10,66,38),(11,67,39),(12,68,40),(13,69,41),(14,70,42),(15,71,43),(16,72,44),(17,73,45),(18,74,46),(19,75,47),(20,76,48),(21,77,49),(22,78,50),(23,79,51),(24,80,52),(25,81,53),(26,82,54),(27,83,55),(28,84,56),(85,141,113),(86,142,114),(87,143,115),(88,144,116),(89,145,117),(90,146,118),(91,147,119),(92,148,120),(93,149,121),(94,150,122),(95,151,123),(96,152,124),(97,153,125),(98,154,126),(99,155,127),(100,156,128),(101,157,129),(102,158,130),(103,159,131),(104,160,132),(105,161,133),(106,162,134),(107,163,135),(108,164,136),(109,165,137),(110,166,138),(111,167,139),(112,168,140)], [(1,22,8,15),(2,23,9,16),(3,24,10,17),(4,25,11,18),(5,26,12,19),(6,27,13,20),(7,28,14,21),(29,50,36,43),(30,51,37,44),(31,52,38,45),(32,53,39,46),(33,54,40,47),(34,55,41,48),(35,56,42,49),(57,78,64,71),(58,79,65,72),(59,80,66,73),(60,81,67,74),(61,82,68,75),(62,83,69,76),(63,84,70,77),(85,99,92,106),(86,100,93,107),(87,101,94,108),(88,102,95,109),(89,103,96,110),(90,104,97,111),(91,105,98,112),(113,127,120,134),(114,128,121,135),(115,129,122,136),(116,130,123,137),(117,131,124,138),(118,132,125,139),(119,133,126,140),(141,155,148,162),(142,156,149,163),(143,157,150,164),(144,158,151,165),(145,159,152,166),(146,160,153,167),(147,161,154,168)], [(1,106),(2,107),(3,108),(4,109),(5,110),(6,111),(7,112),(8,99),(9,100),(10,101),(11,102),(12,103),(13,104),(14,105),(15,85),(16,86),(17,87),(18,88),(19,89),(20,90),(21,91),(22,92),(23,93),(24,94),(25,95),(26,96),(27,97),(28,98),(29,134),(30,135),(31,136),(32,137),(33,138),(34,139),(35,140),(36,127),(37,128),(38,129),(39,130),(40,131),(41,132),(42,133),(43,113),(44,114),(45,115),(46,116),(47,117),(48,118),(49,119),(50,120),(51,121),(52,122),(53,123),(54,124),(55,125),(56,126),(57,162),(58,163),(59,164),(60,165),(61,166),(62,167),(63,168),(64,155),(65,156),(66,157),(67,158),(68,159),(69,160),(70,161),(71,141),(72,142),(73,143),(74,144),(75,145),(76,146),(77,147),(78,148),(79,149),(80,150),(81,151),(82,152),(83,153),(84,154)], [(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)], [(1,7),(2,6),(3,5),(8,14),(9,13),(10,12),(15,28),(16,27),(17,26),(18,25),(19,24),(20,23),(21,22),(29,35),(30,34),(31,33),(36,42),(37,41),(38,40),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50),(57,63),(58,62),(59,61),(64,70),(65,69),(66,68),(71,84),(72,83),(73,82),(74,81),(75,80),(76,79),(77,78),(85,105),(86,104),(87,103),(88,102),(89,101),(90,100),(91,99),(92,112),(93,111),(94,110),(95,109),(96,108),(97,107),(98,106),(113,133),(114,132),(115,131),(116,130),(117,129),(118,128),(119,127),(120,140),(121,139),(122,138),(123,137),(124,136),(125,135),(126,134),(141,161),(142,160),(143,159),(144,158),(145,157),(146,156),(147,155),(148,168),(149,167),(150,166),(151,165),(152,164),(153,163),(154,162)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4 | 6A | 6B | 6C | 6D | 6E | 6F | 7A | 7B | 7C | 8A | 8B | 12A | 12B | 14A | 14B | 14C | 14D | ··· | 14I | 21A | ··· | 21F | 24A | 24B | 24C | 24D | 28A | 28B | 28C | 42A | ··· | 42F | 42G | ··· | 42R | 84A | ··· | 84F |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 8 | 8 | 12 | 12 | 14 | 14 | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 24 | 24 | 24 | 24 | 28 | 28 | 28 | 42 | ··· | 42 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 4 | 28 | 1 | 1 | 2 | 1 | 1 | 4 | 4 | 28 | 28 | 2 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 14 | 14 | 14 | 14 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D7 | D8 | C3×D4 | D14 | C3×D7 | C3×D8 | C7⋊D4 | C6×D7 | C3×C7⋊D4 | D4⋊D7 | C3×D4⋊D7 |
| kernel | C3×D4⋊D7 | C3×C7⋊C8 | C3×D28 | D4×C21 | D4⋊D7 | C7⋊C8 | D28 | C7×D4 | C42 | C3×D4 | C21 | C14 | C12 | D4 | C7 | C6 | C4 | C2 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 3 | 2 | 2 | 3 | 6 | 4 | 6 | 6 | 12 | 3 | 6 |
Matrix representation of C3×D4⋊D7 ►in GL4(𝔽337) generated by
| 208 | 0 | 0 | 0 |
| 0 | 208 | 0 | 0 |
| 0 | 0 | 128 | 0 |
| 0 | 0 | 0 | 128 |
| 336 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 |
| 0 | 0 | 1 | 141 |
| 0 | 0 | 141 | 336 |
| 242 | 274 | 0 | 0 |
| 63 | 95 | 0 | 0 |
| 0 | 0 | 311 | 189 |
| 0 | 0 | 189 | 26 |
| 227 | 336 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 227 | 336 | 0 | 0 |
| 304 | 110 | 0 | 0 |
| 0 | 0 | 1 | 141 |
| 0 | 0 | 0 | 336 |
G:=sub<GL(4,GF(337))| [208,0,0,0,0,208,0,0,0,0,128,0,0,0,0,128],[336,0,0,0,0,336,0,0,0,0,1,141,0,0,141,336],[242,63,0,0,274,95,0,0,0,0,311,189,0,0,189,26],[227,1,0,0,336,0,0,0,0,0,1,0,0,0,0,1],[227,304,0,0,336,110,0,0,0,0,1,0,0,0,141,336] >;
C3×D4⋊D7 in GAP, Magma, Sage, TeX
C_3\times D_4\rtimes D_7
% in TeX
G:=Group("C3xD4:D7"); // GroupNames label
G:=SmallGroup(336,69);
// by ID
G=gap.SmallGroup(336,69);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,169,867,441,69,10373]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations
Export