direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic20, C10⋊1Q16, C4.8D20, C8.16D10, C20.31D4, C20.31C23, C40.18C22, C22.14D20, Dic10.7C22, C5⋊1(C2×Q16), (C2×C8).4D5, (C2×C40).6C2, C2.14(C2×D20), C10.12(C2×D4), (C2×C4).82D10, (C2×C10).19D4, C4.29(C22×D5), (C2×C20).90C22, (C2×Dic10).5C2, SmallGroup(160,126)
Series: Derived ►Chief ►Lower central ►Upper central
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
►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)]])
C2×Dic20 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 C2×D5×Q16 D20.47D4
C2×Dic20 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 | 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