direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C41, C4⋊C82, C22⋊C82, C164⋊3C2, C82.6C22, (C2×C82)⋊1C2, C2.1(C2×C82), SmallGroup(328,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C41
G = < a,b,c | a41=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×C41
►On 164 points
Generators in S164
(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)
(1 155 73 123)(2 156 74 83)(3 157 75 84)(4 158 76 85)(5 159 77 86)(6 160 78 87)(7 161 79 88)(8 162 80 89)(9 163 81 90)(10 164 82 91)(11 124 42 92)(12 125 43 93)(13 126 44 94)(14 127 45 95)(15 128 46 96)(16 129 47 97)(17 130 48 98)(18 131 49 99)(19 132 50 100)(20 133 51 101)(21 134 52 102)(22 135 53 103)(23 136 54 104)(24 137 55 105)(25 138 56 106)(26 139 57 107)(27 140 58 108)(28 141 59 109)(29 142 60 110)(30 143 61 111)(31 144 62 112)(32 145 63 113)(33 146 64 114)(34 147 65 115)(35 148 66 116)(36 149 67 117)(37 150 68 118)(38 151 69 119)(39 152 70 120)(40 153 71 121)(41 154 72 122)
(83 156)(84 157)(85 158)(86 159)(87 160)(88 161)(89 162)(90 163)(91 164)(92 124)(93 125)(94 126)(95 127)(96 128)(97 129)(98 130)(99 131)(100 132)(101 133)(102 134)(103 135)(104 136)(105 137)(106 138)(107 139)(108 140)(109 141)(110 142)(111 143)(112 144)(113 145)(114 146)(115 147)(116 148)(117 149)(118 150)(119 151)(120 152)(121 153)(122 154)(123 155)
G:=sub<Sym(164)| (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), (1,155,73,123)(2,156,74,83)(3,157,75,84)(4,158,76,85)(5,159,77,86)(6,160,78,87)(7,161,79,88)(8,162,80,89)(9,163,81,90)(10,164,82,91)(11,124,42,92)(12,125,43,93)(13,126,44,94)(14,127,45,95)(15,128,46,96)(16,129,47,97)(17,130,48,98)(18,131,49,99)(19,132,50,100)(20,133,51,101)(21,134,52,102)(22,135,53,103)(23,136,54,104)(24,137,55,105)(25,138,56,106)(26,139,57,107)(27,140,58,108)(28,141,59,109)(29,142,60,110)(30,143,61,111)(31,144,62,112)(32,145,63,113)(33,146,64,114)(34,147,65,115)(35,148,66,116)(36,149,67,117)(37,150,68,118)(38,151,69,119)(39,152,70,120)(40,153,71,121)(41,154,72,122), (83,156)(84,157)(85,158)(86,159)(87,160)(88,161)(89,162)(90,163)(91,164)(92,124)(93,125)(94,126)(95,127)(96,128)(97,129)(98,130)(99,131)(100,132)(101,133)(102,134)(103,135)(104,136)(105,137)(106,138)(107,139)(108,140)(109,141)(110,142)(111,143)(112,144)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152)(121,153)(122,154)(123,155)>;
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), (1,155,73,123)(2,156,74,83)(3,157,75,84)(4,158,76,85)(5,159,77,86)(6,160,78,87)(7,161,79,88)(8,162,80,89)(9,163,81,90)(10,164,82,91)(11,124,42,92)(12,125,43,93)(13,126,44,94)(14,127,45,95)(15,128,46,96)(16,129,47,97)(17,130,48,98)(18,131,49,99)(19,132,50,100)(20,133,51,101)(21,134,52,102)(22,135,53,103)(23,136,54,104)(24,137,55,105)(25,138,56,106)(26,139,57,107)(27,140,58,108)(28,141,59,109)(29,142,60,110)(30,143,61,111)(31,144,62,112)(32,145,63,113)(33,146,64,114)(34,147,65,115)(35,148,66,116)(36,149,67,117)(37,150,68,118)(38,151,69,119)(39,152,70,120)(40,153,71,121)(41,154,72,122), (83,156)(84,157)(85,158)(86,159)(87,160)(88,161)(89,162)(90,163)(91,164)(92,124)(93,125)(94,126)(95,127)(96,128)(97,129)(98,130)(99,131)(100,132)(101,133)(102,134)(103,135)(104,136)(105,137)(106,138)(107,139)(108,140)(109,141)(110,142)(111,143)(112,144)(113,145)(114,146)(115,147)(116,148)(117,149)(118,150)(119,151)(120,152)(121,153)(122,154)(123,155) );
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)], [(1,155,73,123),(2,156,74,83),(3,157,75,84),(4,158,76,85),(5,159,77,86),(6,160,78,87),(7,161,79,88),(8,162,80,89),(9,163,81,90),(10,164,82,91),(11,124,42,92),(12,125,43,93),(13,126,44,94),(14,127,45,95),(15,128,46,96),(16,129,47,97),(17,130,48,98),(18,131,49,99),(19,132,50,100),(20,133,51,101),(21,134,52,102),(22,135,53,103),(23,136,54,104),(24,137,55,105),(25,138,56,106),(26,139,57,107),(27,140,58,108),(28,141,59,109),(29,142,60,110),(30,143,61,111),(31,144,62,112),(32,145,63,113),(33,146,64,114),(34,147,65,115),(35,148,66,116),(36,149,67,117),(37,150,68,118),(38,151,69,119),(39,152,70,120),(40,153,71,121),(41,154,72,122)], [(83,156),(84,157),(85,158),(86,159),(87,160),(88,161),(89,162),(90,163),(91,164),(92,124),(93,125),(94,126),(95,127),(96,128),(97,129),(98,130),(99,131),(100,132),(101,133),(102,134),(103,135),(104,136),(105,137),(106,138),(107,139),(108,140),(109,141),(110,142),(111,143),(112,144),(113,145),(114,146),(115,147),(116,148),(117,149),(118,150),(119,151),(120,152),(121,153),(122,154),(123,155)]])
205 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 41A | ··· | 41AN | 82A | ··· | 82AN | 82AO | ··· | 82DP | 164A | ··· | 164AN |
| order | 1 | 2 | 2 | 2 | 4 | 41 | ··· | 41 | 82 | ··· | 82 | 82 | ··· | 82 | 164 | ··· | 164 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
205 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C41 | C82 | C82 | D4 | D4×C41 |
| kernel | D4×C41 | C164 | C2×C82 | D4 | C4 | C22 | C41 | C1 |
| # reps | 1 | 1 | 2 | 40 | 40 | 80 | 1 | 40 |
Matrix representation of D4×C41 ►in GL2(𝔽821) generated by
| 132 | 0 |
| 0 | 132 |
| 0 | 820 |
| 1 | 0 |
| 1 | 0 |
| 0 | 820 |
G:=sub<GL(2,GF(821))| [132,0,0,132],[0,1,820,0],[1,0,0,820] >;
D4×C41 in GAP, Magma, Sage, TeX
D_4\times C_{41} % in TeX
G:=Group("D4xC41"); // GroupNames label
G:=SmallGroup(328,10);
// by ID
G=gap.SmallGroup(328,10);
# by ID
G:=PCGroup([4,-2,-2,-41,-2,1329]);
// Polycyclic
G:=Group<a,b,c|a^41=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export