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G = C22×C40order 160 = 25·5

Abelian group of type [2,2,40]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C40, SmallGroup(160,190)

Series: Derived Chief Lower central Upper central

C1 — C22×C40
C1C2C4C20C40C2×C40 — C22×C40
C1 — C22×C40
C1 — C22×C40

Generators and relations for C22×C40
 G = < a,b,c | a2=b2=c40=1, ab=ba, ac=ca, bc=cb >

Subgroups: 76, all normal (12 characteristic)
C1, C2, C2 [×6], C4, C4 [×3], C22 [×7], C5, C8 [×4], C2×C4 [×6], C23, C10, C10 [×6], C2×C8 [×6], C22×C4, C20, C20 [×3], C2×C10 [×7], C22×C8, C40 [×4], C2×C20 [×6], C22×C10, C2×C40 [×6], C22×C20, C22×C40
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C8 [×4], C2×C4 [×6], C23, C10 [×7], C2×C8 [×6], C22×C4, C20 [×4], C2×C10 [×7], C22×C8, C40 [×4], C2×C20 [×6], C22×C10, C2×C40 [×6], C22×C20, C22×C40

Smallest permutation representation of C22×C40
Regular action on 160 points
Generators in S160
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 73)(21 74)(22 75)(23 76)(24 77)(25 78)(26 79)(27 80)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 52)(40 53)(81 129)(82 130)(83 131)(84 132)(85 133)(86 134)(87 135)(88 136)(89 137)(90 138)(91 139)(92 140)(93 141)(94 142)(95 143)(96 144)(97 145)(98 146)(99 147)(100 148)(101 149)(102 150)(103 151)(104 152)(105 153)(106 154)(107 155)(108 156)(109 157)(110 158)(111 159)(112 160)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 127)(120 128)
(1 144)(2 145)(3 146)(4 147)(5 148)(6 149)(7 150)(8 151)(9 152)(10 153)(11 154)(12 155)(13 156)(14 157)(15 158)(16 159)(17 160)(18 121)(19 122)(20 123)(21 124)(22 125)(23 126)(24 127)(25 128)(26 129)(27 130)(28 131)(29 132)(30 133)(31 134)(32 135)(33 136)(34 137)(35 138)(36 139)(37 140)(38 141)(39 142)(40 143)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 97)(56 98)(57 99)(58 100)(59 101)(60 102)(61 103)(62 104)(63 105)(64 106)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 113)(72 114)(73 115)(74 116)(75 117)(76 118)(77 119)(78 120)(79 81)(80 82)
(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)

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

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

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

C22×C40 is a maximal subgroup of
C40.91D4  (C2×C40)⋊15C4  C20.39C42  C20.40C42  C20.42C42  C20.65(C4⋊C4)  C23.22D20  (C22×C8)⋊D5  C4032D4  C23.23D20  C4030D4  C4029D4  C40.82D4

160 conjugacy classes

class 1 2A···2G4A···4H5A5B5C5D8A···8P10A···10AB20A···20AF40A···40BL
order12···24···455558···810···1020···2040···40
size11···11···111111···11···11···11···1

160 irreducible representations

dim111111111111
type+++
imageC1C2C2C4C4C5C8C10C10C20C20C40
kernelC22×C40C2×C40C22×C20C2×C20C22×C10C22×C8C2×C10C2×C8C22×C4C2×C4C23C22
# reps1616241624424864

Matrix representation of C22×C40 in GL3(𝔽41) generated by

4000
010
0040
,
100
0400
001
,
1900
0220
0018
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,40],[1,0,0,0,40,0,0,0,1],[19,0,0,0,22,0,0,0,18] >;

C22×C40 in GAP, Magma, Sage, TeX

C_2^2\times C_{40}
% in TeX

G:=Group("C2^2xC40");
// GroupNames label

G:=SmallGroup(160,190);
// by ID

G=gap.SmallGroup(160,190);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,240,88]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^40=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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