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G = D4×C41order 328 = 23·41

Direct product of C41 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C41, C4⋊C82, C22⋊C82, C1643C2, C82.6C22, (C2×C82)⋊1C2, C2.1(C2×C82), SmallGroup(328,10)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C41
C1C2C82C2×C82 — D4×C41
C1C2 — D4×C41
C1C82 — D4×C41

Generators and relations for D4×C41
 G = < a,b,c | a41=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C82
2C82

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)
(1 161 61 85)(2 162 62 86)(3 163 63 87)(4 164 64 88)(5 124 65 89)(6 125 66 90)(7 126 67 91)(8 127 68 92)(9 128 69 93)(10 129 70 94)(11 130 71 95)(12 131 72 96)(13 132 73 97)(14 133 74 98)(15 134 75 99)(16 135 76 100)(17 136 77 101)(18 137 78 102)(19 138 79 103)(20 139 80 104)(21 140 81 105)(22 141 82 106)(23 142 42 107)(24 143 43 108)(25 144 44 109)(26 145 45 110)(27 146 46 111)(28 147 47 112)(29 148 48 113)(30 149 49 114)(31 150 50 115)(32 151 51 116)(33 152 52 117)(34 153 53 118)(35 154 54 119)(36 155 55 120)(37 156 56 121)(38 157 57 122)(39 158 58 123)(40 159 59 83)(41 160 60 84)
(83 159)(84 160)(85 161)(86 162)(87 163)(88 164)(89 124)(90 125)(91 126)(92 127)(93 128)(94 129)(95 130)(96 131)(97 132)(98 133)(99 134)(100 135)(101 136)(102 137)(103 138)(104 139)(105 140)(106 141)(107 142)(108 143)(109 144)(110 145)(111 146)(112 147)(113 148)(114 149)(115 150)(116 151)(117 152)(118 153)(119 154)(120 155)(121 156)(122 157)(123 158)

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), (1,161,61,85)(2,162,62,86)(3,163,63,87)(4,164,64,88)(5,124,65,89)(6,125,66,90)(7,126,67,91)(8,127,68,92)(9,128,69,93)(10,129,70,94)(11,130,71,95)(12,131,72,96)(13,132,73,97)(14,133,74,98)(15,134,75,99)(16,135,76,100)(17,136,77,101)(18,137,78,102)(19,138,79,103)(20,139,80,104)(21,140,81,105)(22,141,82,106)(23,142,42,107)(24,143,43,108)(25,144,44,109)(26,145,45,110)(27,146,46,111)(28,147,47,112)(29,148,48,113)(30,149,49,114)(31,150,50,115)(32,151,51,116)(33,152,52,117)(34,153,53,118)(35,154,54,119)(36,155,55,120)(37,156,56,121)(38,157,57,122)(39,158,58,123)(40,159,59,83)(41,160,60,84), (83,159)(84,160)(85,161)(86,162)(87,163)(88,164)(89,124)(90,125)(91,126)(92,127)(93,128)(94,129)(95,130)(96,131)(97,132)(98,133)(99,134)(100,135)(101,136)(102,137)(103,138)(104,139)(105,140)(106,141)(107,142)(108,143)(109,144)(110,145)(111,146)(112,147)(113,148)(114,149)(115,150)(116,151)(117,152)(118,153)(119,154)(120,155)(121,156)(122,157)(123,158)>;

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), (1,161,61,85)(2,162,62,86)(3,163,63,87)(4,164,64,88)(5,124,65,89)(6,125,66,90)(7,126,67,91)(8,127,68,92)(9,128,69,93)(10,129,70,94)(11,130,71,95)(12,131,72,96)(13,132,73,97)(14,133,74,98)(15,134,75,99)(16,135,76,100)(17,136,77,101)(18,137,78,102)(19,138,79,103)(20,139,80,104)(21,140,81,105)(22,141,82,106)(23,142,42,107)(24,143,43,108)(25,144,44,109)(26,145,45,110)(27,146,46,111)(28,147,47,112)(29,148,48,113)(30,149,49,114)(31,150,50,115)(32,151,51,116)(33,152,52,117)(34,153,53,118)(35,154,54,119)(36,155,55,120)(37,156,56,121)(38,157,57,122)(39,158,58,123)(40,159,59,83)(41,160,60,84), (83,159)(84,160)(85,161)(86,162)(87,163)(88,164)(89,124)(90,125)(91,126)(92,127)(93,128)(94,129)(95,130)(96,131)(97,132)(98,133)(99,134)(100,135)(101,136)(102,137)(103,138)(104,139)(105,140)(106,141)(107,142)(108,143)(109,144)(110,145)(111,146)(112,147)(113,148)(114,149)(115,150)(116,151)(117,152)(118,153)(119,154)(120,155)(121,156)(122,157)(123,158) );

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)], [(1,161,61,85),(2,162,62,86),(3,163,63,87),(4,164,64,88),(5,124,65,89),(6,125,66,90),(7,126,67,91),(8,127,68,92),(9,128,69,93),(10,129,70,94),(11,130,71,95),(12,131,72,96),(13,132,73,97),(14,133,74,98),(15,134,75,99),(16,135,76,100),(17,136,77,101),(18,137,78,102),(19,138,79,103),(20,139,80,104),(21,140,81,105),(22,141,82,106),(23,142,42,107),(24,143,43,108),(25,144,44,109),(26,145,45,110),(27,146,46,111),(28,147,47,112),(29,148,48,113),(30,149,49,114),(31,150,50,115),(32,151,51,116),(33,152,52,117),(34,153,53,118),(35,154,54,119),(36,155,55,120),(37,156,56,121),(38,157,57,122),(39,158,58,123),(40,159,59,83),(41,160,60,84)], [(83,159),(84,160),(85,161),(86,162),(87,163),(88,164),(89,124),(90,125),(91,126),(92,127),(93,128),(94,129),(95,130),(96,131),(97,132),(98,133),(99,134),(100,135),(101,136),(102,137),(103,138),(104,139),(105,140),(106,141),(107,142),(108,143),(109,144),(110,145),(111,146),(112,147),(113,148),(114,149),(115,150),(116,151),(117,152),(118,153),(119,154),(120,155),(121,156),(122,157),(123,158)])

205 conjugacy classes

class 1 2A2B2C 4 41A···41AN82A···82AN82AO···82DP164A···164AN
order1222441···4182···8282···82164···164
size112221···11···12···22···2

205 irreducible representations

dim11111122
type++++
imageC1C2C2C41C82C82D4D4×C41
kernelD4×C41C164C2×C82D4C4C22C41C1
# reps112404080140

Matrix representation of D4×C41 in GL2(𝔽821) generated by

1320
0132
,
0820
10
,
10
0820
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

Subgroup lattice of D4×C41 in TeX

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