direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D7×C3⋊C8, C28.29D6, C12.29D14, C84.29C22, D14.2Dic3, Dic7.2Dic3, (C3×D7)⋊C8, C3⋊3(C8×D7), C21⋊1(C2×C8), C21⋊C8⋊9C2, C42.1(C2×C4), (C4×D7).3S3, (C6×D7).1C4, C4.22(S3×D7), C6.10(C4×D7), (C12×D7).2C2, C2.1(Dic3×D7), (C3×Dic7).1C4, C14.1(C2×Dic3), C7⋊1(C2×C3⋊C8), (C7×C3⋊C8)⋊4C2, SmallGroup(336,23)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — D7×C3⋊C8 |
Generators and relations for D7×C3⋊C8
G = < a,b,c,d | a7=b2=c3=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of D7×C3⋊C8
►On 168 points
Generators in S168
(1 165 15 105 119 135 46)(2 166 16 106 120 136 47)(3 167 9 107 113 129 48)(4 168 10 108 114 130 41)(5 161 11 109 115 131 42)(6 162 12 110 116 132 43)(7 163 13 111 117 133 44)(8 164 14 112 118 134 45)(17 40 29 82 147 153 54)(18 33 30 83 148 154 55)(19 34 31 84 149 155 56)(20 35 32 85 150 156 49)(21 36 25 86 151 157 50)(22 37 26 87 152 158 51)(23 38 27 88 145 159 52)(24 39 28 81 146 160 53)(57 79 72 124 139 92 101)(58 80 65 125 140 93 102)(59 73 66 126 141 94 103)(60 74 67 127 142 95 104)(61 75 68 128 143 96 97)(62 76 69 121 144 89 98)(63 77 70 122 137 90 99)(64 78 71 123 138 91 100)
(1 46)(2 47)(3 48)(4 41)(5 42)(6 43)(7 44)(8 45)(9 113)(10 114)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 153)(18 154)(19 155)(20 156)(21 157)(22 158)(23 159)(24 160)(25 86)(26 87)(27 88)(28 81)(29 82)(30 83)(31 84)(32 85)(33 148)(34 149)(35 150)(36 151)(37 152)(38 145)(39 146)(40 147)(57 72)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(64 71)(89 144)(90 137)(91 138)(92 139)(93 140)(94 141)(95 142)(96 143)(97 128)(98 121)(99 122)(100 123)(101 124)(102 125)(103 126)(104 127)(129 167)(130 168)(131 161)(132 162)(133 163)(134 164)(135 165)(136 166)
(1 90 82)(2 83 91)(3 92 84)(4 85 93)(5 94 86)(6 87 95)(7 96 88)(8 81 89)(9 57 155)(10 156 58)(11 59 157)(12 158 60)(13 61 159)(14 160 62)(15 63 153)(16 154 64)(17 119 70)(18 71 120)(19 113 72)(20 65 114)(21 115 66)(22 67 116)(23 117 68)(24 69 118)(25 42 141)(26 142 43)(27 44 143)(28 144 45)(29 46 137)(30 138 47)(31 48 139)(32 140 41)(33 123 136)(34 129 124)(35 125 130)(36 131 126)(37 127 132)(38 133 128)(39 121 134)(40 135 122)(49 80 108)(50 109 73)(51 74 110)(52 111 75)(53 76 112)(54 105 77)(55 78 106)(56 107 79)(97 145 163)(98 164 146)(99 147 165)(100 166 148)(101 149 167)(102 168 150)(103 151 161)(104 162 152)
(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)
G:=sub<Sym(168)| (1,165,15,105,119,135,46)(2,166,16,106,120,136,47)(3,167,9,107,113,129,48)(4,168,10,108,114,130,41)(5,161,11,109,115,131,42)(6,162,12,110,116,132,43)(7,163,13,111,117,133,44)(8,164,14,112,118,134,45)(17,40,29,82,147,153,54)(18,33,30,83,148,154,55)(19,34,31,84,149,155,56)(20,35,32,85,150,156,49)(21,36,25,86,151,157,50)(22,37,26,87,152,158,51)(23,38,27,88,145,159,52)(24,39,28,81,146,160,53)(57,79,72,124,139,92,101)(58,80,65,125,140,93,102)(59,73,66,126,141,94,103)(60,74,67,127,142,95,104)(61,75,68,128,143,96,97)(62,76,69,121,144,89,98)(63,77,70,122,137,90,99)(64,78,71,123,138,91,100), (1,46)(2,47)(3,48)(4,41)(5,42)(6,43)(7,44)(8,45)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,153)(18,154)(19,155)(20,156)(21,157)(22,158)(23,159)(24,160)(25,86)(26,87)(27,88)(28,81)(29,82)(30,83)(31,84)(32,85)(33,148)(34,149)(35,150)(36,151)(37,152)(38,145)(39,146)(40,147)(57,72)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(64,71)(89,144)(90,137)(91,138)(92,139)(93,140)(94,141)(95,142)(96,143)(97,128)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127)(129,167)(130,168)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166), (1,90,82)(2,83,91)(3,92,84)(4,85,93)(5,94,86)(6,87,95)(7,96,88)(8,81,89)(9,57,155)(10,156,58)(11,59,157)(12,158,60)(13,61,159)(14,160,62)(15,63,153)(16,154,64)(17,119,70)(18,71,120)(19,113,72)(20,65,114)(21,115,66)(22,67,116)(23,117,68)(24,69,118)(25,42,141)(26,142,43)(27,44,143)(28,144,45)(29,46,137)(30,138,47)(31,48,139)(32,140,41)(33,123,136)(34,129,124)(35,125,130)(36,131,126)(37,127,132)(38,133,128)(39,121,134)(40,135,122)(49,80,108)(50,109,73)(51,74,110)(52,111,75)(53,76,112)(54,105,77)(55,78,106)(56,107,79)(97,145,163)(98,164,146)(99,147,165)(100,166,148)(101,149,167)(102,168,150)(103,151,161)(104,162,152), (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)>;
G:=Group( (1,165,15,105,119,135,46)(2,166,16,106,120,136,47)(3,167,9,107,113,129,48)(4,168,10,108,114,130,41)(5,161,11,109,115,131,42)(6,162,12,110,116,132,43)(7,163,13,111,117,133,44)(8,164,14,112,118,134,45)(17,40,29,82,147,153,54)(18,33,30,83,148,154,55)(19,34,31,84,149,155,56)(20,35,32,85,150,156,49)(21,36,25,86,151,157,50)(22,37,26,87,152,158,51)(23,38,27,88,145,159,52)(24,39,28,81,146,160,53)(57,79,72,124,139,92,101)(58,80,65,125,140,93,102)(59,73,66,126,141,94,103)(60,74,67,127,142,95,104)(61,75,68,128,143,96,97)(62,76,69,121,144,89,98)(63,77,70,122,137,90,99)(64,78,71,123,138,91,100), (1,46)(2,47)(3,48)(4,41)(5,42)(6,43)(7,44)(8,45)(9,113)(10,114)(11,115)(12,116)(13,117)(14,118)(15,119)(16,120)(17,153)(18,154)(19,155)(20,156)(21,157)(22,158)(23,159)(24,160)(25,86)(26,87)(27,88)(28,81)(29,82)(30,83)(31,84)(32,85)(33,148)(34,149)(35,150)(36,151)(37,152)(38,145)(39,146)(40,147)(57,72)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(64,71)(89,144)(90,137)(91,138)(92,139)(93,140)(94,141)(95,142)(96,143)(97,128)(98,121)(99,122)(100,123)(101,124)(102,125)(103,126)(104,127)(129,167)(130,168)(131,161)(132,162)(133,163)(134,164)(135,165)(136,166), (1,90,82)(2,83,91)(3,92,84)(4,85,93)(5,94,86)(6,87,95)(7,96,88)(8,81,89)(9,57,155)(10,156,58)(11,59,157)(12,158,60)(13,61,159)(14,160,62)(15,63,153)(16,154,64)(17,119,70)(18,71,120)(19,113,72)(20,65,114)(21,115,66)(22,67,116)(23,117,68)(24,69,118)(25,42,141)(26,142,43)(27,44,143)(28,144,45)(29,46,137)(30,138,47)(31,48,139)(32,140,41)(33,123,136)(34,129,124)(35,125,130)(36,131,126)(37,127,132)(38,133,128)(39,121,134)(40,135,122)(49,80,108)(50,109,73)(51,74,110)(52,111,75)(53,76,112)(54,105,77)(55,78,106)(56,107,79)(97,145,163)(98,164,146)(99,147,165)(100,166,148)(101,149,167)(102,168,150)(103,151,161)(104,162,152), (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) );
G=PermutationGroup([[(1,165,15,105,119,135,46),(2,166,16,106,120,136,47),(3,167,9,107,113,129,48),(4,168,10,108,114,130,41),(5,161,11,109,115,131,42),(6,162,12,110,116,132,43),(7,163,13,111,117,133,44),(8,164,14,112,118,134,45),(17,40,29,82,147,153,54),(18,33,30,83,148,154,55),(19,34,31,84,149,155,56),(20,35,32,85,150,156,49),(21,36,25,86,151,157,50),(22,37,26,87,152,158,51),(23,38,27,88,145,159,52),(24,39,28,81,146,160,53),(57,79,72,124,139,92,101),(58,80,65,125,140,93,102),(59,73,66,126,141,94,103),(60,74,67,127,142,95,104),(61,75,68,128,143,96,97),(62,76,69,121,144,89,98),(63,77,70,122,137,90,99),(64,78,71,123,138,91,100)], [(1,46),(2,47),(3,48),(4,41),(5,42),(6,43),(7,44),(8,45),(9,113),(10,114),(11,115),(12,116),(13,117),(14,118),(15,119),(16,120),(17,153),(18,154),(19,155),(20,156),(21,157),(22,158),(23,159),(24,160),(25,86),(26,87),(27,88),(28,81),(29,82),(30,83),(31,84),(32,85),(33,148),(34,149),(35,150),(36,151),(37,152),(38,145),(39,146),(40,147),(57,72),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(64,71),(89,144),(90,137),(91,138),(92,139),(93,140),(94,141),(95,142),(96,143),(97,128),(98,121),(99,122),(100,123),(101,124),(102,125),(103,126),(104,127),(129,167),(130,168),(131,161),(132,162),(133,163),(134,164),(135,165),(136,166)], [(1,90,82),(2,83,91),(3,92,84),(4,85,93),(5,94,86),(6,87,95),(7,96,88),(8,81,89),(9,57,155),(10,156,58),(11,59,157),(12,158,60),(13,61,159),(14,160,62),(15,63,153),(16,154,64),(17,119,70),(18,71,120),(19,113,72),(20,65,114),(21,115,66),(22,67,116),(23,117,68),(24,69,118),(25,42,141),(26,142,43),(27,44,143),(28,144,45),(29,46,137),(30,138,47),(31,48,139),(32,140,41),(33,123,136),(34,129,124),(35,125,130),(36,131,126),(37,127,132),(38,133,128),(39,121,134),(40,135,122),(49,80,108),(50,109,73),(51,74,110),(52,111,75),(53,76,112),(54,105,77),(55,78,106),(56,107,79),(97,145,163),(98,164,146),(99,147,165),(100,166,148),(101,149,167),(102,168,150),(103,151,161),(104,162,152)], [(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)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 14A | 14B | 14C | 21A | 21B | 21C | 28A | ··· | 28F | 42A | 42B | 42C | 56A | ··· | 56L | 84A | ··· | 84F |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 14 | 14 | 14 | 21 | 21 | 21 | 28 | ··· | 28 | 42 | 42 | 42 | 56 | ··· | 56 | 84 | ··· | 84 |
| size | 1 | 1 | 7 | 7 | 2 | 1 | 1 | 7 | 7 | 2 | 14 | 14 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 21 | 21 | 21 | 21 | 2 | 2 | 14 | 14 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | Dic3 | D6 | Dic3 | D7 | C3⋊C8 | D14 | C4×D7 | C8×D7 | S3×D7 | Dic3×D7 | D7×C3⋊C8 |
| kernel | D7×C3⋊C8 | C7×C3⋊C8 | C21⋊C8 | C12×D7 | C3×Dic7 | C6×D7 | C3×D7 | C4×D7 | Dic7 | C28 | D14 | C3⋊C8 | D7 | C12 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 3 | 4 | 3 | 6 | 12 | 3 | 3 | 6 |
Matrix representation of D7×C3⋊C8 ►in GL4(𝔽337) generated by
| 0 | 1 | 0 | 0 |
| 336 | 303 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 336 |
| 0 | 0 | 1 | 336 |
| 336 | 0 | 0 | 0 |
| 0 | 336 | 0 | 0 |
| 0 | 0 | 187 | 287 |
| 0 | 0 | 137 | 150 |
G:=sub<GL(4,GF(337))| [0,336,0,0,1,303,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,336,336],[336,0,0,0,0,336,0,0,0,0,187,137,0,0,287,150] >;
D7×C3⋊C8 in GAP, Magma, Sage, TeX
D_7\times C_3\rtimes C_8
% in TeX
G:=Group("D7xC3:C8"); // GroupNames label
G:=SmallGroup(336,23);
// by ID
G=gap.SmallGroup(336,23);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,31,50,490,10373]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^2=c^3=d^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export