direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×D43, C43⋊C23, C86⋊C22, (C2×C86)⋊3C2, SmallGroup(344,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C43 — C22×D43 |
Generators and relations for C22×D43
G = < a,b,c,d | a2=b2=c43=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of C22×D43
►On 172 points
Generators in S172
(1 142)(2 143)(3 144)(4 145)(5 146)(6 147)(7 148)(8 149)(9 150)(10 151)(11 152)(12 153)(13 154)(14 155)(15 156)(16 157)(17 158)(18 159)(19 160)(20 161)(21 162)(22 163)(23 164)(24 165)(25 166)(26 167)(27 168)(28 169)(29 170)(30 171)(31 172)(32 130)(33 131)(34 132)(35 133)(36 134)(37 135)(38 136)(39 137)(40 138)(41 139)(42 140)(43 141)(44 98)(45 99)(46 100)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)(55 109)(56 110)(57 111)(58 112)(59 113)(60 114)(61 115)(62 116)(63 117)(64 118)(65 119)(66 120)(67 121)(68 122)(69 123)(70 124)(71 125)(72 126)(73 127)(74 128)(75 129)(76 87)(77 88)(78 89)(79 90)(80 91)(81 92)(82 93)(83 94)(84 95)(85 96)(86 97)
(1 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 81)(12 82)(13 83)(14 84)(15 85)(16 86)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(41 68)(42 69)(43 70)(87 147)(88 148)(89 149)(90 150)(91 151)(92 152)(93 153)(94 154)(95 155)(96 156)(97 157)(98 158)(99 159)(100 160)(101 161)(102 162)(103 163)(104 164)(105 165)(106 166)(107 167)(108 168)(109 169)(110 170)(111 171)(112 172)(113 130)(114 131)(115 132)(116 133)(117 134)(118 135)(119 136)(120 137)(121 138)(122 139)(123 140)(124 141)(125 142)(126 143)(127 144)(128 145)(129 146)
(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)
(1 141)(2 140)(3 139)(4 138)(5 137)(6 136)(7 135)(8 134)(9 133)(10 132)(11 131)(12 130)(13 172)(14 171)(15 170)(16 169)(17 168)(18 167)(19 166)(20 165)(21 164)(22 163)(23 162)(24 161)(25 160)(26 159)(27 158)(28 157)(29 156)(30 155)(31 154)(32 153)(33 152)(34 151)(35 150)(36 149)(37 148)(38 147)(39 146)(40 145)(41 144)(42 143)(43 142)(44 108)(45 107)(46 106)(47 105)(48 104)(49 103)(50 102)(51 101)(52 100)(53 99)(54 98)(55 97)(56 96)(57 95)(58 94)(59 93)(60 92)(61 91)(62 90)(63 89)(64 88)(65 87)(66 129)(67 128)(68 127)(69 126)(70 125)(71 124)(72 123)(73 122)(74 121)(75 120)(76 119)(77 118)(78 117)(79 116)(80 115)(81 114)(82 113)(83 112)(84 111)(85 110)(86 109)
G:=sub<Sym(172)| (1,142)(2,143)(3,144)(4,145)(5,146)(6,147)(7,148)(8,149)(9,150)(10,151)(11,152)(12,153)(13,154)(14,155)(15,156)(16,157)(17,158)(18,159)(19,160)(20,161)(21,162)(22,163)(23,164)(24,165)(25,166)(26,167)(27,168)(28,169)(29,170)(30,171)(31,172)(32,130)(33,131)(34,132)(35,133)(36,134)(37,135)(38,136)(39,137)(40,138)(41,139)(42,140)(43,141)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(61,115)(62,116)(63,117)(64,118)(65,119)(66,120)(67,121)(68,122)(69,123)(70,124)(71,125)(72,126)(73,127)(74,128)(75,129)(76,87)(77,88)(78,89)(79,90)(80,91)(81,92)(82,93)(83,94)(84,95)(85,96)(86,97), (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,81)(12,82)(13,83)(14,84)(15,85)(16,86)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(87,147)(88,148)(89,149)(90,150)(91,151)(92,152)(93,153)(94,154)(95,155)(96,156)(97,157)(98,158)(99,159)(100,160)(101,161)(102,162)(103,163)(104,164)(105,165)(106,166)(107,167)(108,168)(109,169)(110,170)(111,171)(112,172)(113,130)(114,131)(115,132)(116,133)(117,134)(118,135)(119,136)(120,137)(121,138)(122,139)(123,140)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146), (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), (1,141)(2,140)(3,139)(4,138)(5,137)(6,136)(7,135)(8,134)(9,133)(10,132)(11,131)(12,130)(13,172)(14,171)(15,170)(16,169)(17,168)(18,167)(19,166)(20,165)(21,164)(22,163)(23,162)(24,161)(25,160)(26,159)(27,158)(28,157)(29,156)(30,155)(31,154)(32,153)(33,152)(34,151)(35,150)(36,149)(37,148)(38,147)(39,146)(40,145)(41,144)(42,143)(43,142)(44,108)(45,107)(46,106)(47,105)(48,104)(49,103)(50,102)(51,101)(52,100)(53,99)(54,98)(55,97)(56,96)(57,95)(58,94)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,87)(66,129)(67,128)(68,127)(69,126)(70,125)(71,124)(72,123)(73,122)(74,121)(75,120)(76,119)(77,118)(78,117)(79,116)(80,115)(81,114)(82,113)(83,112)(84,111)(85,110)(86,109)>;
G:=Group( (1,142)(2,143)(3,144)(4,145)(5,146)(6,147)(7,148)(8,149)(9,150)(10,151)(11,152)(12,153)(13,154)(14,155)(15,156)(16,157)(17,158)(18,159)(19,160)(20,161)(21,162)(22,163)(23,164)(24,165)(25,166)(26,167)(27,168)(28,169)(29,170)(30,171)(31,172)(32,130)(33,131)(34,132)(35,133)(36,134)(37,135)(38,136)(39,137)(40,138)(41,139)(42,140)(43,141)(44,98)(45,99)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(61,115)(62,116)(63,117)(64,118)(65,119)(66,120)(67,121)(68,122)(69,123)(70,124)(71,125)(72,126)(73,127)(74,128)(75,129)(76,87)(77,88)(78,89)(79,90)(80,91)(81,92)(82,93)(83,94)(84,95)(85,96)(86,97), (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,81)(12,82)(13,83)(14,84)(15,85)(16,86)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,61)(35,62)(36,63)(37,64)(38,65)(39,66)(40,67)(41,68)(42,69)(43,70)(87,147)(88,148)(89,149)(90,150)(91,151)(92,152)(93,153)(94,154)(95,155)(96,156)(97,157)(98,158)(99,159)(100,160)(101,161)(102,162)(103,163)(104,164)(105,165)(106,166)(107,167)(108,168)(109,169)(110,170)(111,171)(112,172)(113,130)(114,131)(115,132)(116,133)(117,134)(118,135)(119,136)(120,137)(121,138)(122,139)(123,140)(124,141)(125,142)(126,143)(127,144)(128,145)(129,146), (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), (1,141)(2,140)(3,139)(4,138)(5,137)(6,136)(7,135)(8,134)(9,133)(10,132)(11,131)(12,130)(13,172)(14,171)(15,170)(16,169)(17,168)(18,167)(19,166)(20,165)(21,164)(22,163)(23,162)(24,161)(25,160)(26,159)(27,158)(28,157)(29,156)(30,155)(31,154)(32,153)(33,152)(34,151)(35,150)(36,149)(37,148)(38,147)(39,146)(40,145)(41,144)(42,143)(43,142)(44,108)(45,107)(46,106)(47,105)(48,104)(49,103)(50,102)(51,101)(52,100)(53,99)(54,98)(55,97)(56,96)(57,95)(58,94)(59,93)(60,92)(61,91)(62,90)(63,89)(64,88)(65,87)(66,129)(67,128)(68,127)(69,126)(70,125)(71,124)(72,123)(73,122)(74,121)(75,120)(76,119)(77,118)(78,117)(79,116)(80,115)(81,114)(82,113)(83,112)(84,111)(85,110)(86,109) );
G=PermutationGroup([[(1,142),(2,143),(3,144),(4,145),(5,146),(6,147),(7,148),(8,149),(9,150),(10,151),(11,152),(12,153),(13,154),(14,155),(15,156),(16,157),(17,158),(18,159),(19,160),(20,161),(21,162),(22,163),(23,164),(24,165),(25,166),(26,167),(27,168),(28,169),(29,170),(30,171),(31,172),(32,130),(33,131),(34,132),(35,133),(36,134),(37,135),(38,136),(39,137),(40,138),(41,139),(42,140),(43,141),(44,98),(45,99),(46,100),(47,101),(48,102),(49,103),(50,104),(51,105),(52,106),(53,107),(54,108),(55,109),(56,110),(57,111),(58,112),(59,113),(60,114),(61,115),(62,116),(63,117),(64,118),(65,119),(66,120),(67,121),(68,122),(69,123),(70,124),(71,125),(72,126),(73,127),(74,128),(75,129),(76,87),(77,88),(78,89),(79,90),(80,91),(81,92),(82,93),(83,94),(84,95),(85,96),(86,97)], [(1,71),(2,72),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,79),(10,80),(11,81),(12,82),(13,83),(14,84),(15,85),(16,86),(17,44),(18,45),(19,46),(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,61),(35,62),(36,63),(37,64),(38,65),(39,66),(40,67),(41,68),(42,69),(43,70),(87,147),(88,148),(89,149),(90,150),(91,151),(92,152),(93,153),(94,154),(95,155),(96,156),(97,157),(98,158),(99,159),(100,160),(101,161),(102,162),(103,163),(104,164),(105,165),(106,166),(107,167),(108,168),(109,169),(110,170),(111,171),(112,172),(113,130),(114,131),(115,132),(116,133),(117,134),(118,135),(119,136),(120,137),(121,138),(122,139),(123,140),(124,141),(125,142),(126,143),(127,144),(128,145),(129,146)], [(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)], [(1,141),(2,140),(3,139),(4,138),(5,137),(6,136),(7,135),(8,134),(9,133),(10,132),(11,131),(12,130),(13,172),(14,171),(15,170),(16,169),(17,168),(18,167),(19,166),(20,165),(21,164),(22,163),(23,162),(24,161),(25,160),(26,159),(27,158),(28,157),(29,156),(30,155),(31,154),(32,153),(33,152),(34,151),(35,150),(36,149),(37,148),(38,147),(39,146),(40,145),(41,144),(42,143),(43,142),(44,108),(45,107),(46,106),(47,105),(48,104),(49,103),(50,102),(51,101),(52,100),(53,99),(54,98),(55,97),(56,96),(57,95),(58,94),(59,93),(60,92),(61,91),(62,90),(63,89),(64,88),(65,87),(66,129),(67,128),(68,127),(69,126),(70,125),(71,124),(72,123),(73,122),(74,121),(75,120),(76,119),(77,118),(78,117),(79,116),(80,115),(81,114),(82,113),(83,112),(84,111),(85,110),(86,109)]])
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 43A | ··· | 43U | 86A | ··· | 86BK |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 43 | ··· | 43 | 86 | ··· | 86 |
| size | 1 | 1 | 1 | 1 | 43 | 43 | 43 | 43 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + |
| image | C1 | C2 | C2 | D43 | D86 |
| kernel | C22×D43 | D86 | C2×C86 | C22 | C2 |
| # reps | 1 | 6 | 1 | 21 | 63 |
Matrix representation of C22×D43 ►in GL3(𝔽173) generated by
| 172 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 172 | 0 |
| 0 | 0 | 172 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 172 | 94 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(173))| [172,0,0,0,1,0,0,0,1],[1,0,0,0,172,0,0,0,172],[1,0,0,0,0,172,0,1,94],[1,0,0,0,0,1,0,1,0] >;
C22×D43 in GAP, Magma, Sage, TeX
C_2^2\times D_{43} % in TeX
G:=Group("C2^2xD43"); // GroupNames label
G:=SmallGroup(344,11);
// by ID
G=gap.SmallGroup(344,11);
# by ID
G:=PCGroup([4,-2,-2,-2,-43,5379]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^43=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export