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## G = C2×Dic20order 160 = 25·5

### Direct product of C2 and Dic20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×Dic20
 Chief series C1 — C5 — C10 — C20 — Dic10 — C2×Dic10 — C2×Dic20
 Lower central C5 — C10 — C20 — C2×Dic20
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C2×Dic20
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, C2×C4, C2×C4, Q8, C10, C10, C2×C8, Q16, C2×Q8, Dic5, C20, C2×C10, C2×Q16, C40, Dic10, Dic10, C2×Dic5, C2×C20, Dic20, C2×C40, C2×Dic10, C2×Dic20
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, D10, C2×Q16, D20, C22×D5, Dic20, C2×D20, C2×Dic20

Smallest permutation representation of C2×Dic20
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)]])

46 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A ··· 20H 40A ··· 40P order 1 2 2 2 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 20 20 20 20 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

46 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + - + + + + - image C1 C2 C2 C2 D4 D4 D5 Q16 D10 D10 D20 D20 Dic20 kernel C2×Dic20 Dic20 C2×C40 C2×Dic10 C20 C2×C10 C2×C8 C10 C8 C2×C4 C4 C22 C2 # reps 1 4 1 2 1 1 2 4 4 2 4 4 16

Matrix representation of C2×Dic20 in GL3(𝔽41) generated by

 40 0 0 0 1 0 0 0 1
,
 1 0 0 0 33 36 0 5 3
,
 1 0 0 0 10 37 0 15 31
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] >;

C2×Dic20 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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