direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4×D20, C42⋊38D10, C10⋊2(C4×D4), (C2×C20)⋊32D4, C20⋊11(C2×D4), (C2×C42)⋊7D5, C20⋊8(C22×C4), (C4×C20)⋊53C22, D10⋊2(C22×C4), C2.1(C22×D20), C10.2(C22×D4), C10.25(C23×C4), (C2×C10).17C24, C4⋊Dic5⋊81C22, C22.63(C2×D20), (C2×C20).875C23, (C22×D20).21C2, (C22×C4).468D10, D10⋊C4⋊75C22, C22.14(C23×D5), (C2×D20).292C22, C22.68(C4○D20), C23.314(C22×D5), (C22×C10).379C23, (C22×C20).503C22, (C2×Dic5).183C23, (C22×D5).156C23, (C23×D5).104C22, (C22×Dic5).224C22, C5⋊2(C2×C4×D4), C4⋊3(C2×C4×D5), (C2×C4×C20)⋊11C2, (C2×C4)⋊12(C4×D5), (C2×C20)⋊42(C2×C4), C2.6(D5×C22×C4), C2.3(C2×C4○D20), C10.5(C2×C4○D4), (D5×C22×C4)⋊14C2, (C2×C4×D5)⋊62C22, C22.69(C2×C4×D5), (C2×C4⋊Dic5)⋊49C2, (C2×C10).169(C2×D4), (C22×D5)⋊14(C2×C4), (C2×D10⋊C4)⋊45C2, (C2×C10).96(C4○D4), (C2×C4).817(C22×D5), (C2×C10).248(C22×C4), SmallGroup(320,1145)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4×D20
G = < a,b,c,d | a2=b4=c20=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1518 in 426 conjugacy classes, 183 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C22×D4, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C2×C4×D4, C4⋊Dic5, D10⋊C4, C4×C20, C2×C4×D5, C2×C4×D5, C2×D20, C22×Dic5, C22×C20, C23×D5, C4×D20, C2×C4⋊Dic5, C2×D10⋊C4, C2×C4×C20, D5×C22×C4, C22×D20, C2×C4×D20
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22×C4, C2×D4, C4○D4, C24, D10, C4×D4, C23×C4, C22×D4, C2×C4○D4, C4×D5, D20, C22×D5, C2×C4×D4, C2×C4×D5, C2×D20, C4○D20, C23×D5, C4×D20, D5×C22×C4, C22×D20, C2×C4○D20, C2×C4×D20
►On 160 points
Generators in S160
(1 116)(2 117)(3 118)(4 119)(5 120)(6 101)(7 102)(8 103)(9 104)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 111)(17 112)(18 113)(19 114)(20 115)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 68)(40 69)(41 142)(42 143)(43 144)(44 145)(45 146)(46 147)(47 148)(48 149)(49 150)(50 151)(51 152)(52 153)(53 154)(54 155)(55 156)(56 157)(57 158)(58 159)(59 160)(60 141)(81 133)(82 134)(83 135)(84 136)(85 137)(86 138)(87 139)(88 140)(89 121)(90 122)(91 123)(92 124)(93 125)(94 126)(95 127)(96 128)(97 129)(98 130)(99 131)(100 132)
(1 32 50 87)(2 33 51 88)(3 34 52 89)(4 35 53 90)(5 36 54 91)(6 37 55 92)(7 38 56 93)(8 39 57 94)(9 40 58 95)(10 21 59 96)(11 22 60 97)(12 23 41 98)(13 24 42 99)(14 25 43 100)(15 26 44 81)(16 27 45 82)(17 28 46 83)(18 29 47 84)(19 30 48 85)(20 31 49 86)(61 151 139 116)(62 152 140 117)(63 153 121 118)(64 154 122 119)(65 155 123 120)(66 156 124 101)(67 157 125 102)(68 158 126 103)(69 159 127 104)(70 160 128 105)(71 141 129 106)(72 142 130 107)(73 143 131 108)(74 144 132 109)(75 145 133 110)(76 146 134 111)(77 147 135 112)(78 148 136 113)(79 149 137 114)(80 150 138 115)
(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)
(1 105)(2 104)(3 103)(4 102)(5 101)(6 120)(7 119)(8 118)(9 117)(10 116)(11 115)(12 114)(13 113)(14 112)(15 111)(16 110)(17 109)(18 108)(19 107)(20 106)(21 61)(22 80)(23 79)(24 78)(25 77)(26 76)(27 75)(28 74)(29 73)(30 72)(31 71)(32 70)(33 69)(34 68)(35 67)(36 66)(37 65)(38 64)(39 63)(40 62)(41 149)(42 148)(43 147)(44 146)(45 145)(46 144)(47 143)(48 142)(49 141)(50 160)(51 159)(52 158)(53 157)(54 156)(55 155)(56 154)(57 153)(58 152)(59 151)(60 150)(81 134)(82 133)(83 132)(84 131)(85 130)(86 129)(87 128)(88 127)(89 126)(90 125)(91 124)(92 123)(93 122)(94 121)(95 140)(96 139)(97 138)(98 137)(99 136)(100 135)
G:=sub<Sym(160)| (1,116)(2,117)(3,118)(4,119)(5,120)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(41,142)(42,143)(43,144)(44,145)(45,146)(46,147)(47,148)(48,149)(49,150)(50,151)(51,152)(52,153)(53,154)(54,155)(55,156)(56,157)(57,158)(58,159)(59,160)(60,141)(81,133)(82,134)(83,135)(84,136)(85,137)(86,138)(87,139)(88,140)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128)(97,129)(98,130)(99,131)(100,132), (1,32,50,87)(2,33,51,88)(3,34,52,89)(4,35,53,90)(5,36,54,91)(6,37,55,92)(7,38,56,93)(8,39,57,94)(9,40,58,95)(10,21,59,96)(11,22,60,97)(12,23,41,98)(13,24,42,99)(14,25,43,100)(15,26,44,81)(16,27,45,82)(17,28,46,83)(18,29,47,84)(19,30,48,85)(20,31,49,86)(61,151,139,116)(62,152,140,117)(63,153,121,118)(64,154,122,119)(65,155,123,120)(66,156,124,101)(67,157,125,102)(68,158,126,103)(69,159,127,104)(70,160,128,105)(71,141,129,106)(72,142,130,107)(73,143,131,108)(74,144,132,109)(75,145,133,110)(76,146,134,111)(77,147,135,112)(78,148,136,113)(79,149,137,114)(80,150,138,115), (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), (1,105)(2,104)(3,103)(4,102)(5,101)(6,120)(7,119)(8,118)(9,117)(10,116)(11,115)(12,114)(13,113)(14,112)(15,111)(16,110)(17,109)(18,108)(19,107)(20,106)(21,61)(22,80)(23,79)(24,78)(25,77)(26,76)(27,75)(28,74)(29,73)(30,72)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62)(41,149)(42,148)(43,147)(44,146)(45,145)(46,144)(47,143)(48,142)(49,141)(50,160)(51,159)(52,158)(53,157)(54,156)(55,155)(56,154)(57,153)(58,152)(59,151)(60,150)(81,134)(82,133)(83,132)(84,131)(85,130)(86,129)(87,128)(88,127)(89,126)(90,125)(91,124)(92,123)(93,122)(94,121)(95,140)(96,139)(97,138)(98,137)(99,136)(100,135)>;
G:=Group( (1,116)(2,117)(3,118)(4,119)(5,120)(6,101)(7,102)(8,103)(9,104)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,61)(33,62)(34,63)(35,64)(36,65)(37,66)(38,67)(39,68)(40,69)(41,142)(42,143)(43,144)(44,145)(45,146)(46,147)(47,148)(48,149)(49,150)(50,151)(51,152)(52,153)(53,154)(54,155)(55,156)(56,157)(57,158)(58,159)(59,160)(60,141)(81,133)(82,134)(83,135)(84,136)(85,137)(86,138)(87,139)(88,140)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128)(97,129)(98,130)(99,131)(100,132), (1,32,50,87)(2,33,51,88)(3,34,52,89)(4,35,53,90)(5,36,54,91)(6,37,55,92)(7,38,56,93)(8,39,57,94)(9,40,58,95)(10,21,59,96)(11,22,60,97)(12,23,41,98)(13,24,42,99)(14,25,43,100)(15,26,44,81)(16,27,45,82)(17,28,46,83)(18,29,47,84)(19,30,48,85)(20,31,49,86)(61,151,139,116)(62,152,140,117)(63,153,121,118)(64,154,122,119)(65,155,123,120)(66,156,124,101)(67,157,125,102)(68,158,126,103)(69,159,127,104)(70,160,128,105)(71,141,129,106)(72,142,130,107)(73,143,131,108)(74,144,132,109)(75,145,133,110)(76,146,134,111)(77,147,135,112)(78,148,136,113)(79,149,137,114)(80,150,138,115), (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), (1,105)(2,104)(3,103)(4,102)(5,101)(6,120)(7,119)(8,118)(9,117)(10,116)(11,115)(12,114)(13,113)(14,112)(15,111)(16,110)(17,109)(18,108)(19,107)(20,106)(21,61)(22,80)(23,79)(24,78)(25,77)(26,76)(27,75)(28,74)(29,73)(30,72)(31,71)(32,70)(33,69)(34,68)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62)(41,149)(42,148)(43,147)(44,146)(45,145)(46,144)(47,143)(48,142)(49,141)(50,160)(51,159)(52,158)(53,157)(54,156)(55,155)(56,154)(57,153)(58,152)(59,151)(60,150)(81,134)(82,133)(83,132)(84,131)(85,130)(86,129)(87,128)(88,127)(89,126)(90,125)(91,124)(92,123)(93,122)(94,121)(95,140)(96,139)(97,138)(98,137)(99,136)(100,135) );
G=PermutationGroup([[(1,116),(2,117),(3,118),(4,119),(5,120),(6,101),(7,102),(8,103),(9,104),(10,105),(11,106),(12,107),(13,108),(14,109),(15,110),(16,111),(17,112),(18,113),(19,114),(20,115),(21,70),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,61),(33,62),(34,63),(35,64),(36,65),(37,66),(38,67),(39,68),(40,69),(41,142),(42,143),(43,144),(44,145),(45,146),(46,147),(47,148),(48,149),(49,150),(50,151),(51,152),(52,153),(53,154),(54,155),(55,156),(56,157),(57,158),(58,159),(59,160),(60,141),(81,133),(82,134),(83,135),(84,136),(85,137),(86,138),(87,139),(88,140),(89,121),(90,122),(91,123),(92,124),(93,125),(94,126),(95,127),(96,128),(97,129),(98,130),(99,131),(100,132)], [(1,32,50,87),(2,33,51,88),(3,34,52,89),(4,35,53,90),(5,36,54,91),(6,37,55,92),(7,38,56,93),(8,39,57,94),(9,40,58,95),(10,21,59,96),(11,22,60,97),(12,23,41,98),(13,24,42,99),(14,25,43,100),(15,26,44,81),(16,27,45,82),(17,28,46,83),(18,29,47,84),(19,30,48,85),(20,31,49,86),(61,151,139,116),(62,152,140,117),(63,153,121,118),(64,154,122,119),(65,155,123,120),(66,156,124,101),(67,157,125,102),(68,158,126,103),(69,159,127,104),(70,160,128,105),(71,141,129,106),(72,142,130,107),(73,143,131,108),(74,144,132,109),(75,145,133,110),(76,146,134,111),(77,147,135,112),(78,148,136,113),(79,149,137,114),(80,150,138,115)], [(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)], [(1,105),(2,104),(3,103),(4,102),(5,101),(6,120),(7,119),(8,118),(9,117),(10,116),(11,115),(12,114),(13,113),(14,112),(15,111),(16,110),(17,109),(18,108),(19,107),(20,106),(21,61),(22,80),(23,79),(24,78),(25,77),(26,76),(27,75),(28,74),(29,73),(30,72),(31,71),(32,70),(33,69),(34,68),(35,67),(36,66),(37,65),(38,64),(39,63),(40,62),(41,149),(42,148),(43,147),(44,146),(45,145),(46,144),(47,143),(48,142),(49,141),(50,160),(51,159),(52,158),(53,157),(54,156),(55,155),(56,154),(57,153),(58,152),(59,151),(60,150),(81,134),(82,133),(83,132),(84,131),(85,130),(86,129),(87,128),(88,127),(89,126),(90,125),(91,124),(92,123),(93,122),(94,121),(95,140),(96,139),(97,138),(98,137),(99,136),(100,135)]])
104 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4H | 4I | ··· | 4P | 4Q | ··· | 4X | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20AV |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 10 | ··· | 10 | 1 | ··· | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | C4○D4 | D10 | D10 | C4×D5 | D20 | C4○D20 |
| kernel | C2×C4×D20 | C4×D20 | C2×C4⋊Dic5 | C2×D10⋊C4 | C2×C4×C20 | D5×C22×C4 | C22×D20 | C2×D20 | C2×C20 | C2×C42 | C2×C10 | C42 | C22×C4 | C2×C4 | C2×C4 | C22 |
| # reps | 1 | 8 | 1 | 2 | 1 | 2 | 1 | 16 | 4 | 2 | 4 | 8 | 6 | 16 | 16 | 16 |
Matrix representation of C2×C4×D20 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 5 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 40 | 0 | 0 |
| 0 | 0 | 36 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 0 | 0 | 0 | 0 | 30 | 14 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 36 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 5 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 11 |
| 0 | 0 | 0 | 0 | 30 | 32 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,36,0,0,0,0,40,6,0,0,0,0,0,0,9,30,0,0,0,0,11,14],[1,36,0,0,0,0,0,40,0,0,0,0,0,0,40,5,0,0,0,0,0,1,0,0,0,0,0,0,9,30,0,0,0,0,11,32] >;
C2×C4×D20 in GAP, Magma, Sage, TeX
C_2\times C_4\times D_{20} % in TeX
G:=Group("C2xC4xD20"); // GroupNames label
G:=SmallGroup(320,1145);
// by ID
G=gap.SmallGroup(320,1145);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^20=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