direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×D14⋊C4, D14⋊3C12, C42.27D4, C6.17D28, (C2×C28)⋊9C6, (C2×C84)⋊1C2, (C6×D7)⋊3C4, (C2×C12)⋊1D7, C2.2(C3×D28), C6.19(C4×D7), C2.5(C12×D7), C21⋊5(C22⋊C4), C42.23(C2×C4), (C6×Dic7)⋊7C2, (C2×Dic7)⋊7C6, C14.22(C3×D4), (C2×C6).34D14, C22.6(C6×D7), C14.18(C2×C12), C6.22(C7⋊D4), (C22×D7).3C6, (C2×C42).35C22, (C2×C4)⋊1(C3×D7), C7⋊4(C3×C22⋊C4), (C2×C6×D7).3C2, C2.2(C3×C7⋊D4), (C2×C14).23(C2×C6), SmallGroup(336,68)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D14⋊C4
G = < a,b,c,d | a3=b14=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b7c >
Subgroups: 272 in 68 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C7, C2×C4, C2×C4, C23, C12, C2×C6, C2×C6, D7, C14, C22⋊C4, C21, C2×C12, C2×C12, C22×C6, Dic7, C28, D14, D14, C2×C14, C3×D7, C42, C3×C22⋊C4, C2×Dic7, C2×C28, C22×D7, C3×Dic7, C84, C6×D7, C6×D7, C2×C42, D14⋊C4, C6×Dic7, C2×C84, C2×C6×D7, C3×D14⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, D7, C22⋊C4, C2×C12, C3×D4, D14, C3×D7, C3×C22⋊C4, C4×D7, D28, C7⋊D4, C6×D7, D14⋊C4, C12×D7, C3×D28, C3×C7⋊D4, C3×D14⋊C4
►On 168 points
Generators in S168
(1 66 29)(2 67 30)(3 68 31)(4 69 32)(5 70 33)(6 57 34)(7 58 35)(8 59 36)(9 60 37)(10 61 38)(11 62 39)(12 63 40)(13 64 41)(14 65 42)(15 72 43)(16 73 44)(17 74 45)(18 75 46)(19 76 47)(20 77 48)(21 78 49)(22 79 50)(23 80 51)(24 81 52)(25 82 53)(26 83 54)(27 84 55)(28 71 56)(85 142 115)(86 143 116)(87 144 117)(88 145 118)(89 146 119)(90 147 120)(91 148 121)(92 149 122)(93 150 123)(94 151 124)(95 152 125)(96 153 126)(97 154 113)(98 141 114)(99 167 136)(100 168 137)(101 155 138)(102 156 139)(103 157 140)(104 158 127)(105 159 128)(106 160 129)(107 161 130)(108 162 131)(109 163 132)(110 164 133)(111 165 134)(112 166 135)
(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 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 18)(16 17)(19 28)(20 27)(21 26)(22 25)(23 24)(29 42)(30 41)(31 40)(32 39)(33 38)(34 37)(35 36)(43 46)(44 45)(47 56)(48 55)(49 54)(50 53)(51 52)(57 60)(58 59)(61 70)(62 69)(63 68)(64 67)(65 66)(71 76)(72 75)(73 74)(77 84)(78 83)(79 82)(80 81)(85 93)(86 92)(87 91)(88 90)(94 98)(95 97)(99 105)(100 104)(101 103)(106 112)(107 111)(108 110)(113 125)(114 124)(115 123)(116 122)(117 121)(118 120)(127 137)(128 136)(129 135)(130 134)(131 133)(138 140)(141 151)(142 150)(143 149)(144 148)(145 147)(152 154)(155 157)(158 168)(159 167)(160 166)(161 165)(162 164)
(1 99 24 93)(2 100 25 94)(3 101 26 95)(4 102 27 96)(5 103 28 97)(6 104 15 98)(7 105 16 85)(8 106 17 86)(9 107 18 87)(10 108 19 88)(11 109 20 89)(12 110 21 90)(13 111 22 91)(14 112 23 92)(29 136 52 123)(30 137 53 124)(31 138 54 125)(32 139 55 126)(33 140 56 113)(34 127 43 114)(35 128 44 115)(36 129 45 116)(37 130 46 117)(38 131 47 118)(39 132 48 119)(40 133 49 120)(41 134 50 121)(42 135 51 122)(57 158 72 141)(58 159 73 142)(59 160 74 143)(60 161 75 144)(61 162 76 145)(62 163 77 146)(63 164 78 147)(64 165 79 148)(65 166 80 149)(66 167 81 150)(67 168 82 151)(68 155 83 152)(69 156 84 153)(70 157 71 154)
G:=sub<Sym(168)| (1,66,29)(2,67,30)(3,68,31)(4,69,32)(5,70,33)(6,57,34)(7,58,35)(8,59,36)(9,60,37)(10,61,38)(11,62,39)(12,63,40)(13,64,41)(14,65,42)(15,72,43)(16,73,44)(17,74,45)(18,75,46)(19,76,47)(20,77,48)(21,78,49)(22,79,50)(23,80,51)(24,81,52)(25,82,53)(26,83,54)(27,84,55)(28,71,56)(85,142,115)(86,143,116)(87,144,117)(88,145,118)(89,146,119)(90,147,120)(91,148,121)(92,149,122)(93,150,123)(94,151,124)(95,152,125)(96,153,126)(97,154,113)(98,141,114)(99,167,136)(100,168,137)(101,155,138)(102,156,139)(103,157,140)(104,158,127)(105,159,128)(106,160,129)(107,161,130)(108,162,131)(109,163,132)(110,164,133)(111,165,134)(112,166,135), (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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,18)(16,17)(19,28)(20,27)(21,26)(22,25)(23,24)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(43,46)(44,45)(47,56)(48,55)(49,54)(50,53)(51,52)(57,60)(58,59)(61,70)(62,69)(63,68)(64,67)(65,66)(71,76)(72,75)(73,74)(77,84)(78,83)(79,82)(80,81)(85,93)(86,92)(87,91)(88,90)(94,98)(95,97)(99,105)(100,104)(101,103)(106,112)(107,111)(108,110)(113,125)(114,124)(115,123)(116,122)(117,121)(118,120)(127,137)(128,136)(129,135)(130,134)(131,133)(138,140)(141,151)(142,150)(143,149)(144,148)(145,147)(152,154)(155,157)(158,168)(159,167)(160,166)(161,165)(162,164), (1,99,24,93)(2,100,25,94)(3,101,26,95)(4,102,27,96)(5,103,28,97)(6,104,15,98)(7,105,16,85)(8,106,17,86)(9,107,18,87)(10,108,19,88)(11,109,20,89)(12,110,21,90)(13,111,22,91)(14,112,23,92)(29,136,52,123)(30,137,53,124)(31,138,54,125)(32,139,55,126)(33,140,56,113)(34,127,43,114)(35,128,44,115)(36,129,45,116)(37,130,46,117)(38,131,47,118)(39,132,48,119)(40,133,49,120)(41,134,50,121)(42,135,51,122)(57,158,72,141)(58,159,73,142)(59,160,74,143)(60,161,75,144)(61,162,76,145)(62,163,77,146)(63,164,78,147)(64,165,79,148)(65,166,80,149)(66,167,81,150)(67,168,82,151)(68,155,83,152)(69,156,84,153)(70,157,71,154)>;
G:=Group( (1,66,29)(2,67,30)(3,68,31)(4,69,32)(5,70,33)(6,57,34)(7,58,35)(8,59,36)(9,60,37)(10,61,38)(11,62,39)(12,63,40)(13,64,41)(14,65,42)(15,72,43)(16,73,44)(17,74,45)(18,75,46)(19,76,47)(20,77,48)(21,78,49)(22,79,50)(23,80,51)(24,81,52)(25,82,53)(26,83,54)(27,84,55)(28,71,56)(85,142,115)(86,143,116)(87,144,117)(88,145,118)(89,146,119)(90,147,120)(91,148,121)(92,149,122)(93,150,123)(94,151,124)(95,152,125)(96,153,126)(97,154,113)(98,141,114)(99,167,136)(100,168,137)(101,155,138)(102,156,139)(103,157,140)(104,158,127)(105,159,128)(106,160,129)(107,161,130)(108,162,131)(109,163,132)(110,164,133)(111,165,134)(112,166,135), (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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,18)(16,17)(19,28)(20,27)(21,26)(22,25)(23,24)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(43,46)(44,45)(47,56)(48,55)(49,54)(50,53)(51,52)(57,60)(58,59)(61,70)(62,69)(63,68)(64,67)(65,66)(71,76)(72,75)(73,74)(77,84)(78,83)(79,82)(80,81)(85,93)(86,92)(87,91)(88,90)(94,98)(95,97)(99,105)(100,104)(101,103)(106,112)(107,111)(108,110)(113,125)(114,124)(115,123)(116,122)(117,121)(118,120)(127,137)(128,136)(129,135)(130,134)(131,133)(138,140)(141,151)(142,150)(143,149)(144,148)(145,147)(152,154)(155,157)(158,168)(159,167)(160,166)(161,165)(162,164), (1,99,24,93)(2,100,25,94)(3,101,26,95)(4,102,27,96)(5,103,28,97)(6,104,15,98)(7,105,16,85)(8,106,17,86)(9,107,18,87)(10,108,19,88)(11,109,20,89)(12,110,21,90)(13,111,22,91)(14,112,23,92)(29,136,52,123)(30,137,53,124)(31,138,54,125)(32,139,55,126)(33,140,56,113)(34,127,43,114)(35,128,44,115)(36,129,45,116)(37,130,46,117)(38,131,47,118)(39,132,48,119)(40,133,49,120)(41,134,50,121)(42,135,51,122)(57,158,72,141)(58,159,73,142)(59,160,74,143)(60,161,75,144)(61,162,76,145)(62,163,77,146)(63,164,78,147)(64,165,79,148)(65,166,80,149)(66,167,81,150)(67,168,82,151)(68,155,83,152)(69,156,84,153)(70,157,71,154) );
G=PermutationGroup([[(1,66,29),(2,67,30),(3,68,31),(4,69,32),(5,70,33),(6,57,34),(7,58,35),(8,59,36),(9,60,37),(10,61,38),(11,62,39),(12,63,40),(13,64,41),(14,65,42),(15,72,43),(16,73,44),(17,74,45),(18,75,46),(19,76,47),(20,77,48),(21,78,49),(22,79,50),(23,80,51),(24,81,52),(25,82,53),(26,83,54),(27,84,55),(28,71,56),(85,142,115),(86,143,116),(87,144,117),(88,145,118),(89,146,119),(90,147,120),(91,148,121),(92,149,122),(93,150,123),(94,151,124),(95,152,125),(96,153,126),(97,154,113),(98,141,114),(99,167,136),(100,168,137),(101,155,138),(102,156,139),(103,157,140),(104,158,127),(105,159,128),(106,160,129),(107,161,130),(108,162,131),(109,163,132),(110,164,133),(111,165,134),(112,166,135)], [(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)], [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,18),(16,17),(19,28),(20,27),(21,26),(22,25),(23,24),(29,42),(30,41),(31,40),(32,39),(33,38),(34,37),(35,36),(43,46),(44,45),(47,56),(48,55),(49,54),(50,53),(51,52),(57,60),(58,59),(61,70),(62,69),(63,68),(64,67),(65,66),(71,76),(72,75),(73,74),(77,84),(78,83),(79,82),(80,81),(85,93),(86,92),(87,91),(88,90),(94,98),(95,97),(99,105),(100,104),(101,103),(106,112),(107,111),(108,110),(113,125),(114,124),(115,123),(116,122),(117,121),(118,120),(127,137),(128,136),(129,135),(130,134),(131,133),(138,140),(141,151),(142,150),(143,149),(144,148),(145,147),(152,154),(155,157),(158,168),(159,167),(160,166),(161,165),(162,164)], [(1,99,24,93),(2,100,25,94),(3,101,26,95),(4,102,27,96),(5,103,28,97),(6,104,15,98),(7,105,16,85),(8,106,17,86),(9,107,18,87),(10,108,19,88),(11,109,20,89),(12,110,21,90),(13,111,22,91),(14,112,23,92),(29,136,52,123),(30,137,53,124),(31,138,54,125),(32,139,55,126),(33,140,56,113),(34,127,43,114),(35,128,44,115),(36,129,45,116),(37,130,46,117),(38,131,47,118),(39,132,48,119),(40,133,49,120),(41,134,50,121),(42,135,51,122),(57,158,72,141),(58,159,73,142),(59,160,74,143),(60,161,75,144),(61,162,76,145),(62,163,77,146),(63,164,78,147),(64,165,79,148),(65,166,80,149),(66,167,81,150),(67,168,82,151),(68,155,83,152),(69,156,84,153),(70,157,71,154)]])
102 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 7A | 7B | 7C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 14A | ··· | 14I | 21A | ··· | 21F | 28A | ··· | 28L | 42A | ··· | 42R | 84A | ··· | 84X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 7 | 7 | 7 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 1 | 1 | 2 | 2 | 14 | 14 | 1 | ··· | 1 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | D7 | C3×D4 | D14 | C3×D7 | C4×D7 | D28 | C7⋊D4 | C6×D7 | C12×D7 | C3×D28 | C3×C7⋊D4 |
| kernel | C3×D14⋊C4 | C6×Dic7 | C2×C84 | C2×C6×D7 | D14⋊C4 | C6×D7 | C2×Dic7 | C2×C28 | C22×D7 | D14 | C42 | C2×C12 | C14 | C2×C6 | C2×C4 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 2 | 3 | 4 | 3 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 |
Matrix representation of C3×D14⋊C4 ►in GL3(𝔽337) generated by
| 128 | 0 | 0 |
| 0 | 128 | 0 |
| 0 | 0 | 128 |
| 1 | 0 | 0 |
| 0 | 143 | 109 |
| 0 | 228 | 228 |
| 336 | 0 | 0 |
| 0 | 143 | 109 |
| 0 | 1 | 194 |
| 189 | 0 | 0 |
| 0 | 305 | 90 |
| 0 | 247 | 32 |
G:=sub<GL(3,GF(337))| [128,0,0,0,128,0,0,0,128],[1,0,0,0,143,228,0,109,228],[336,0,0,0,143,1,0,109,194],[189,0,0,0,305,247,0,90,32] >;
C3×D14⋊C4 in GAP, Magma, Sage, TeX
C_3\times D_{14}\rtimes C_4 % in TeX
G:=Group("C3xD14:C4"); // GroupNames label
G:=SmallGroup(336,68);
// by ID
G=gap.SmallGroup(336,68);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,313,79,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^14=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^7*c>;
// generators/relations