metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C43⋊2D4, C22⋊D43, D86⋊2C2, Dic43⋊C2, C2.5D86, C86.5C22, (C2×C86)⋊2C2, SmallGroup(344,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C43⋊D4
G = < a,b,c | a43=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C43⋊D4
►On 172 points
Generators in S172
(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)(130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172)
(1 94 74 144)(2 93 75 143)(3 92 76 142)(4 91 77 141)(5 90 78 140)(6 89 79 139)(7 88 80 138)(8 87 81 137)(9 129 82 136)(10 128 83 135)(11 127 84 134)(12 126 85 133)(13 125 86 132)(14 124 44 131)(15 123 45 130)(16 122 46 172)(17 121 47 171)(18 120 48 170)(19 119 49 169)(20 118 50 168)(21 117 51 167)(22 116 52 166)(23 115 53 165)(24 114 54 164)(25 113 55 163)(26 112 56 162)(27 111 57 161)(28 110 58 160)(29 109 59 159)(30 108 60 158)(31 107 61 157)(32 106 62 156)(33 105 63 155)(34 104 64 154)(35 103 65 153)(36 102 66 152)(37 101 67 151)(38 100 68 150)(39 99 69 149)(40 98 70 148)(41 97 71 147)(42 96 72 146)(43 95 73 145)
(2 43)(3 42)(4 41)(5 40)(6 39)(7 38)(8 37)(9 36)(10 35)(11 34)(12 33)(13 32)(14 31)(15 30)(16 29)(17 28)(18 27)(19 26)(20 25)(21 24)(22 23)(44 61)(45 60)(46 59)(47 58)(48 57)(49 56)(50 55)(51 54)(52 53)(62 86)(63 85)(64 84)(65 83)(66 82)(67 81)(68 80)(69 79)(70 78)(71 77)(72 76)(73 75)(87 151)(88 150)(89 149)(90 148)(91 147)(92 146)(93 145)(94 144)(95 143)(96 142)(97 141)(98 140)(99 139)(100 138)(101 137)(102 136)(103 135)(104 134)(105 133)(106 132)(107 131)(108 130)(109 172)(110 171)(111 170)(112 169)(113 168)(114 167)(115 166)(116 165)(117 164)(118 163)(119 162)(120 161)(121 160)(122 159)(123 158)(124 157)(125 156)(126 155)(127 154)(128 153)(129 152)
G:=sub<Sym(172)| (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)(130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172), (1,94,74,144)(2,93,75,143)(3,92,76,142)(4,91,77,141)(5,90,78,140)(6,89,79,139)(7,88,80,138)(8,87,81,137)(9,129,82,136)(10,128,83,135)(11,127,84,134)(12,126,85,133)(13,125,86,132)(14,124,44,131)(15,123,45,130)(16,122,46,172)(17,121,47,171)(18,120,48,170)(19,119,49,169)(20,118,50,168)(21,117,51,167)(22,116,52,166)(23,115,53,165)(24,114,54,164)(25,113,55,163)(26,112,56,162)(27,111,57,161)(28,110,58,160)(29,109,59,159)(30,108,60,158)(31,107,61,157)(32,106,62,156)(33,105,63,155)(34,104,64,154)(35,103,65,153)(36,102,66,152)(37,101,67,151)(38,100,68,150)(39,99,69,149)(40,98,70,148)(41,97,71,147)(42,96,72,146)(43,95,73,145), (2,43)(3,42)(4,41)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)(62,86)(63,85)(64,84)(65,83)(66,82)(67,81)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(87,151)(88,150)(89,149)(90,148)(91,147)(92,146)(93,145)(94,144)(95,143)(96,142)(97,141)(98,140)(99,139)(100,138)(101,137)(102,136)(103,135)(104,134)(105,133)(106,132)(107,131)(108,130)(109,172)(110,171)(111,170)(112,169)(113,168)(114,167)(115,166)(116,165)(117,164)(118,163)(119,162)(120,161)(121,160)(122,159)(123,158)(124,157)(125,156)(126,155)(127,154)(128,153)(129,152)>;
G:=Group( (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)(130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172), (1,94,74,144)(2,93,75,143)(3,92,76,142)(4,91,77,141)(5,90,78,140)(6,89,79,139)(7,88,80,138)(8,87,81,137)(9,129,82,136)(10,128,83,135)(11,127,84,134)(12,126,85,133)(13,125,86,132)(14,124,44,131)(15,123,45,130)(16,122,46,172)(17,121,47,171)(18,120,48,170)(19,119,49,169)(20,118,50,168)(21,117,51,167)(22,116,52,166)(23,115,53,165)(24,114,54,164)(25,113,55,163)(26,112,56,162)(27,111,57,161)(28,110,58,160)(29,109,59,159)(30,108,60,158)(31,107,61,157)(32,106,62,156)(33,105,63,155)(34,104,64,154)(35,103,65,153)(36,102,66,152)(37,101,67,151)(38,100,68,150)(39,99,69,149)(40,98,70,148)(41,97,71,147)(42,96,72,146)(43,95,73,145), (2,43)(3,42)(4,41)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)(62,86)(63,85)(64,84)(65,83)(66,82)(67,81)(68,80)(69,79)(70,78)(71,77)(72,76)(73,75)(87,151)(88,150)(89,149)(90,148)(91,147)(92,146)(93,145)(94,144)(95,143)(96,142)(97,141)(98,140)(99,139)(100,138)(101,137)(102,136)(103,135)(104,134)(105,133)(106,132)(107,131)(108,130)(109,172)(110,171)(111,170)(112,169)(113,168)(114,167)(115,166)(116,165)(117,164)(118,163)(119,162)(120,161)(121,160)(122,159)(123,158)(124,157)(125,156)(126,155)(127,154)(128,153)(129,152) );
G=PermutationGroup([[(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),(130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172)], [(1,94,74,144),(2,93,75,143),(3,92,76,142),(4,91,77,141),(5,90,78,140),(6,89,79,139),(7,88,80,138),(8,87,81,137),(9,129,82,136),(10,128,83,135),(11,127,84,134),(12,126,85,133),(13,125,86,132),(14,124,44,131),(15,123,45,130),(16,122,46,172),(17,121,47,171),(18,120,48,170),(19,119,49,169),(20,118,50,168),(21,117,51,167),(22,116,52,166),(23,115,53,165),(24,114,54,164),(25,113,55,163),(26,112,56,162),(27,111,57,161),(28,110,58,160),(29,109,59,159),(30,108,60,158),(31,107,61,157),(32,106,62,156),(33,105,63,155),(34,104,64,154),(35,103,65,153),(36,102,66,152),(37,101,67,151),(38,100,68,150),(39,99,69,149),(40,98,70,148),(41,97,71,147),(42,96,72,146),(43,95,73,145)], [(2,43),(3,42),(4,41),(5,40),(6,39),(7,38),(8,37),(9,36),(10,35),(11,34),(12,33),(13,32),(14,31),(15,30),(16,29),(17,28),(18,27),(19,26),(20,25),(21,24),(22,23),(44,61),(45,60),(46,59),(47,58),(48,57),(49,56),(50,55),(51,54),(52,53),(62,86),(63,85),(64,84),(65,83),(66,82),(67,81),(68,80),(69,79),(70,78),(71,77),(72,76),(73,75),(87,151),(88,150),(89,149),(90,148),(91,147),(92,146),(93,145),(94,144),(95,143),(96,142),(97,141),(98,140),(99,139),(100,138),(101,137),(102,136),(103,135),(104,134),(105,133),(106,132),(107,131),(108,130),(109,172),(110,171),(111,170),(112,169),(113,168),(114,167),(115,166),(116,165),(117,164),(118,163),(119,162),(120,161),(121,160),(122,159),(123,158),(124,157),(125,156),(126,155),(127,154),(128,153),(129,152)]])
89 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 43A | ··· | 43U | 86A | ··· | 86BK |
| order | 1 | 2 | 2 | 2 | 4 | 43 | ··· | 43 | 86 | ··· | 86 |
| size | 1 | 1 | 2 | 86 | 86 | 2 | ··· | 2 | 2 | ··· | 2 |
89 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D43 | D86 | C43⋊D4 |
| kernel | C43⋊D4 | Dic43 | D86 | C2×C86 | C43 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 21 | 21 | 42 |
Matrix representation of C43⋊D4 ►in GL2(𝔽173) generated by
| 157 | 1 |
| 157 | 55 |
| 126 | 146 |
| 37 | 47 |
| 10 | 98 |
| 29 | 163 |
G:=sub<GL(2,GF(173))| [157,157,1,55],[126,37,146,47],[10,29,98,163] >;
C43⋊D4 in GAP, Magma, Sage, TeX
C_{43}\rtimes D_4 % in TeX
G:=Group("C43:D4"); // GroupNames label
G:=SmallGroup(344,7);
// by ID
G=gap.SmallGroup(344,7);
# by ID
G:=PCGroup([4,-2,-2,-2,-43,49,5379]);
// Polycyclic
G:=Group<a,b,c|a^43=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export