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## G = C22×D41order 328 = 23·41

### Direct product of C22 and D41

Aliases: C22×D41, C41⋊C23, C82⋊C22, (C2×C82)⋊3C2, SmallGroup(328,14)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C41 — C22×D41
 Chief series C1 — C41 — D41 — D82 — C22×D41
 Lower central C41 — C22×D41
 Upper central C1 — C22

Generators and relations for C22×D41
G = < a,b,c,d | a2=b2=c41=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Smallest permutation representation of C22×D41
On 164 points
Generators in S164
(1 164)(2 124)(3 125)(4 126)(5 127)(6 128)(7 129)(8 130)(9 131)(10 132)(11 133)(12 134)(13 135)(14 136)(15 137)(16 138)(17 139)(18 140)(19 141)(20 142)(21 143)(22 144)(23 145)(24 146)(25 147)(26 148)(27 149)(28 150)(29 151)(30 152)(31 153)(32 154)(33 155)(34 156)(35 157)(36 158)(37 159)(38 160)(39 161)(40 162)(41 163)(42 86)(43 87)(44 88)(45 89)(46 90)(47 91)(48 92)(49 93)(50 94)(51 95)(52 96)(53 97)(54 98)(55 99)(56 100)(57 101)(58 102)(59 103)(60 104)(61 105)(62 106)(63 107)(64 108)(65 109)(66 110)(67 111)(68 112)(69 113)(70 114)(71 115)(72 116)(73 117)(74 118)(75 119)(76 120)(77 121)(78 122)(79 123)(80 83)(81 84)(82 85)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 81)(32 82)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 49)(41 50)(83 152)(84 153)(85 154)(86 155)(87 156)(88 157)(89 158)(90 159)(91 160)(92 161)(93 162)(94 163)(95 164)(96 124)(97 125)(98 126)(99 127)(100 128)(101 129)(102 130)(103 131)(104 132)(105 133)(106 134)(107 135)(108 136)(109 137)(110 138)(111 139)(112 140)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 149)(122 150)(123 151)
(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)
(1 163)(2 162)(3 161)(4 160)(5 159)(6 158)(7 157)(8 156)(9 155)(10 154)(11 153)(12 152)(13 151)(14 150)(15 149)(16 148)(17 147)(18 146)(19 145)(20 144)(21 143)(22 142)(23 141)(24 140)(25 139)(26 138)(27 137)(28 136)(29 135)(30 134)(31 133)(32 132)(33 131)(34 130)(35 129)(36 128)(37 127)(38 126)(39 125)(40 124)(41 164)(42 103)(43 102)(44 101)(45 100)(46 99)(47 98)(48 97)(49 96)(50 95)(51 94)(52 93)(53 92)(54 91)(55 90)(56 89)(57 88)(58 87)(59 86)(60 85)(61 84)(62 83)(63 123)(64 122)(65 121)(66 120)(67 119)(68 118)(69 117)(70 116)(71 115)(72 114)(73 113)(74 112)(75 111)(76 110)(77 109)(78 108)(79 107)(80 106)(81 105)(82 104)

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

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

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

88 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 41A ··· 41T 82A ··· 82BH order 1 2 2 2 2 2 2 2 41 ··· 41 82 ··· 82 size 1 1 1 1 41 41 41 41 2 ··· 2 2 ··· 2

88 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 D41 D82 kernel C22×D41 D82 C2×C82 C22 C2 # reps 1 6 1 20 60

Matrix representation of C22×D41 in GL3(𝔽83) generated by

 1 0 0 0 82 0 0 0 82
,
 82 0 0 0 82 0 0 0 82
,
 1 0 0 0 82 1 0 50 32
,
 82 0 0 0 1 0 0 33 82
G:=sub<GL(3,GF(83))| [1,0,0,0,82,0,0,0,82],[82,0,0,0,82,0,0,0,82],[1,0,0,0,82,50,0,1,32],[82,0,0,0,1,33,0,0,82] >;

C22×D41 in GAP, Magma, Sage, TeX

C_2^2\times D_{41}
% in TeX

G:=Group("C2^2xD41");
// GroupNames label

G:=SmallGroup(328,14);
// by ID

G=gap.SmallGroup(328,14);
# by ID

G:=PCGroup([4,-2,-2,-2,-41,5123]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^41=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

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