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G = C2xDic20order 160 = 25·5

Direct product of C2 and Dic20

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2xDic20, C10:1Q16, C4.8D20, C8.16D10, C20.31D4, C20.31C23, C40.18C22, C22.14D20, Dic10.7C22, C5:1(C2xQ16), (C2xC8).4D5, (C2xC40).6C2, C2.14(C2xD20), C10.12(C2xD4), (C2xC4).82D10, (C2xC10).19D4, C4.29(C22xD5), (C2xC20).90C22, (C2xDic10).5C2, SmallGroup(160,126)

Series: Derived Chief Lower central Upper central

C1C20 — C2xDic20
C1C5C10C20Dic10C2xDic10 — C2xDic20
C5C10C20 — C2xDic20
C1C22C2xC4C2xC8

Generators and relations for C2xDic20
 G = < a,b,c | a2=b40=1, c2=b20, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 184 in 60 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2xC4, C2xC4, Q8, C10, C10, C2xC8, Q16, C2xQ8, Dic5, C20, C2xC10, C2xQ16, C40, Dic10, Dic10, C2xDic5, C2xC20, Dic20, C2xC40, C2xDic10, C2xDic20
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2xD4, D10, C2xQ16, D20, C22xD5, Dic20, C2xD20, C2xDic20

Smallest permutation representation of C2xDic20
Regular action on 160 points
Generators in S160
(1 80)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(81 133)(82 134)(83 135)(84 136)(85 137)(86 138)(87 139)(88 140)(89 141)(90 142)(91 143)(92 144)(93 145)(94 146)(95 147)(96 148)(97 149)(98 150)(99 151)(100 152)(101 153)(102 154)(103 155)(104 156)(105 157)(106 158)(107 159)(108 160)(109 121)(110 122)(111 123)(112 124)(113 125)(114 126)(115 127)(116 128)(117 129)(118 130)(119 131)(120 132)
(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)
(1 81 21 101)(2 120 22 100)(3 119 23 99)(4 118 24 98)(5 117 25 97)(6 116 26 96)(7 115 27 95)(8 114 28 94)(9 113 29 93)(10 112 30 92)(11 111 31 91)(12 110 32 90)(13 109 33 89)(14 108 34 88)(15 107 35 87)(16 106 36 86)(17 105 37 85)(18 104 38 84)(19 103 39 83)(20 102 40 82)(41 132 61 152)(42 131 62 151)(43 130 63 150)(44 129 64 149)(45 128 65 148)(46 127 66 147)(47 126 67 146)(48 125 68 145)(49 124 69 144)(50 123 70 143)(51 122 71 142)(52 121 72 141)(53 160 73 140)(54 159 74 139)(55 158 75 138)(56 157 76 137)(57 156 77 136)(58 155 78 135)(59 154 79 134)(60 153 80 133)

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

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

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

C2xDic20 is a maximal subgroup of
C10.Q32  C40.8D4  C40.78D4  C20.4D8  C8.8D20  C20:4Q16  C8.D20  Dic20:9C4  D20.32D4  C22:Dic20  Dic10.D4  D4.D20  Dic5:Q16  D10:4Q16  C42.36D10  C4:Dic20  Dic20:15C4  C8.2D20  Dic5:5Q16  D10:2Q16  C8.20D20  C16.D10  C40.82D4  C40.4D4  D4.5D20  C40.22D4  C40.31D4  C40.26D4  C40.31C23  D4.13D20  C2xD5xQ16  D20.47D4
C2xDic20 is a maximal quotient of
C20.14Q16  C40:8Q8  C20:4Q16  C23.35D20  C22:Dic20  C4:Dic20  C20.7Q16  C40.82D4

46 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F5A5B8A8B8C8D10A···10F20A···20H40A···40P
order122244444455888810···1020···2040···40
size111122202020202222222···22···22···2

46 irreducible representations

dim1111222222222
type+++++++-++++-
imageC1C2C2C2D4D4D5Q16D10D10D20D20Dic20
kernelC2xDic20Dic20C2xC40C2xDic10C20C2xC10C2xC8C10C8C2xC4C4C22C2
# reps14121124424416

Matrix representation of C2xDic20 in GL3(F41) generated by

4000
010
001
,
100
03336
053
,
100
01037
01531
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,1],[1,0,0,0,33,5,0,36,3],[1,0,0,0,10,15,0,37,31] >;

C2xDic20 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_{20}
% in TeX

G:=Group("C2xDic20");
// GroupNames label

G:=SmallGroup(160,126);
// by ID

G=gap.SmallGroup(160,126);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,218,122,579,69,4613]);
// Polycyclic

G:=Group<a,b,c|a^2=b^40=1,c^2=b^20,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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