direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C43, C4⋊C86, C22⋊C86, C172⋊3C2, C86.6C22, (C2×C86)⋊1C2, C2.1(C2×C86), SmallGroup(344,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C43
G = < a,b,c | a43=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×C43
►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)
(1 151 90 85)(2 152 91 86)(3 153 92 44)(4 154 93 45)(5 155 94 46)(6 156 95 47)(7 157 96 48)(8 158 97 49)(9 159 98 50)(10 160 99 51)(11 161 100 52)(12 162 101 53)(13 163 102 54)(14 164 103 55)(15 165 104 56)(16 166 105 57)(17 167 106 58)(18 168 107 59)(19 169 108 60)(20 170 109 61)(21 171 110 62)(22 172 111 63)(23 130 112 64)(24 131 113 65)(25 132 114 66)(26 133 115 67)(27 134 116 68)(28 135 117 69)(29 136 118 70)(30 137 119 71)(31 138 120 72)(32 139 121 73)(33 140 122 74)(34 141 123 75)(35 142 124 76)(36 143 125 77)(37 144 126 78)(38 145 127 79)(39 146 128 80)(40 147 129 81)(41 148 87 82)(42 149 88 83)(43 150 89 84)
(44 153)(45 154)(46 155)(47 156)(48 157)(49 158)(50 159)(51 160)(52 161)(53 162)(54 163)(55 164)(56 165)(57 166)(58 167)(59 168)(60 169)(61 170)(62 171)(63 172)(64 130)(65 131)(66 132)(67 133)(68 134)(69 135)(70 136)(71 137)(72 138)(73 139)(74 140)(75 141)(76 142)(77 143)(78 144)(79 145)(80 146)(81 147)(82 148)(83 149)(84 150)(85 151)(86 152)
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), (1,151,90,85)(2,152,91,86)(3,153,92,44)(4,154,93,45)(5,155,94,46)(6,156,95,47)(7,157,96,48)(8,158,97,49)(9,159,98,50)(10,160,99,51)(11,161,100,52)(12,162,101,53)(13,163,102,54)(14,164,103,55)(15,165,104,56)(16,166,105,57)(17,167,106,58)(18,168,107,59)(19,169,108,60)(20,170,109,61)(21,171,110,62)(22,172,111,63)(23,130,112,64)(24,131,113,65)(25,132,114,66)(26,133,115,67)(27,134,116,68)(28,135,117,69)(29,136,118,70)(30,137,119,71)(31,138,120,72)(32,139,121,73)(33,140,122,74)(34,141,123,75)(35,142,124,76)(36,143,125,77)(37,144,126,78)(38,145,127,79)(39,146,128,80)(40,147,129,81)(41,148,87,82)(42,149,88,83)(43,150,89,84), (44,153)(45,154)(46,155)(47,156)(48,157)(49,158)(50,159)(51,160)(52,161)(53,162)(54,163)(55,164)(56,165)(57,166)(58,167)(59,168)(60,169)(61,170)(62,171)(63,172)(64,130)(65,131)(66,132)(67,133)(68,134)(69,135)(70,136)(71,137)(72,138)(73,139)(74,140)(75,141)(76,142)(77,143)(78,144)(79,145)(80,146)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)>;
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), (1,151,90,85)(2,152,91,86)(3,153,92,44)(4,154,93,45)(5,155,94,46)(6,156,95,47)(7,157,96,48)(8,158,97,49)(9,159,98,50)(10,160,99,51)(11,161,100,52)(12,162,101,53)(13,163,102,54)(14,164,103,55)(15,165,104,56)(16,166,105,57)(17,167,106,58)(18,168,107,59)(19,169,108,60)(20,170,109,61)(21,171,110,62)(22,172,111,63)(23,130,112,64)(24,131,113,65)(25,132,114,66)(26,133,115,67)(27,134,116,68)(28,135,117,69)(29,136,118,70)(30,137,119,71)(31,138,120,72)(32,139,121,73)(33,140,122,74)(34,141,123,75)(35,142,124,76)(36,143,125,77)(37,144,126,78)(38,145,127,79)(39,146,128,80)(40,147,129,81)(41,148,87,82)(42,149,88,83)(43,150,89,84), (44,153)(45,154)(46,155)(47,156)(48,157)(49,158)(50,159)(51,160)(52,161)(53,162)(54,163)(55,164)(56,165)(57,166)(58,167)(59,168)(60,169)(61,170)(62,171)(63,172)(64,130)(65,131)(66,132)(67,133)(68,134)(69,135)(70,136)(71,137)(72,138)(73,139)(74,140)(75,141)(76,142)(77,143)(78,144)(79,145)(80,146)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152) );
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)], [(1,151,90,85),(2,152,91,86),(3,153,92,44),(4,154,93,45),(5,155,94,46),(6,156,95,47),(7,157,96,48),(8,158,97,49),(9,159,98,50),(10,160,99,51),(11,161,100,52),(12,162,101,53),(13,163,102,54),(14,164,103,55),(15,165,104,56),(16,166,105,57),(17,167,106,58),(18,168,107,59),(19,169,108,60),(20,170,109,61),(21,171,110,62),(22,172,111,63),(23,130,112,64),(24,131,113,65),(25,132,114,66),(26,133,115,67),(27,134,116,68),(28,135,117,69),(29,136,118,70),(30,137,119,71),(31,138,120,72),(32,139,121,73),(33,140,122,74),(34,141,123,75),(35,142,124,76),(36,143,125,77),(37,144,126,78),(38,145,127,79),(39,146,128,80),(40,147,129,81),(41,148,87,82),(42,149,88,83),(43,150,89,84)], [(44,153),(45,154),(46,155),(47,156),(48,157),(49,158),(50,159),(51,160),(52,161),(53,162),(54,163),(55,164),(56,165),(57,166),(58,167),(59,168),(60,169),(61,170),(62,171),(63,172),(64,130),(65,131),(66,132),(67,133),(68,134),(69,135),(70,136),(71,137),(72,138),(73,139),(74,140),(75,141),(76,142),(77,143),(78,144),(79,145),(80,146),(81,147),(82,148),(83,149),(84,150),(85,151),(86,152)]])
215 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 43A | ··· | 43AP | 86A | ··· | 86AP | 86AQ | ··· | 86DV | 172A | ··· | 172AP |
| order | 1 | 2 | 2 | 2 | 4 | 43 | ··· | 43 | 86 | ··· | 86 | 86 | ··· | 86 | 172 | ··· | 172 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
215 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C43 | C86 | C86 | D4 | D4×C43 |
| kernel | D4×C43 | C172 | C2×C86 | D4 | C4 | C22 | C43 | C1 |
| # reps | 1 | 1 | 2 | 42 | 42 | 84 | 1 | 42 |
Matrix representation of D4×C43 ►in GL2(𝔽173) generated by
| 117 | 0 |
| 0 | 117 |
| 1 | 1 |
| 171 | 172 |
| 1 | 1 |
| 0 | 172 |
G:=sub<GL(2,GF(173))| [117,0,0,117],[1,171,1,172],[1,0,1,172] >;
D4×C43 in GAP, Magma, Sage, TeX
D_4\times C_{43} % in TeX
G:=Group("D4xC43"); // GroupNames label
G:=SmallGroup(344,9);
// by ID
G=gap.SmallGroup(344,9);
# by ID
G:=PCGroup([4,-2,-2,-43,-2,1393]);
// Polycyclic
G:=Group<a,b,c|a^43=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export