direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D20.3C4, C40.71C23, C20.67C24, (C2×C8)⋊37D10, C10⋊3(C8○D4), (C22×C8)⋊13D5, (C22×C40)⋊19C2, (C2×C40)⋊49C22, C4○D20.11C4, (C2×D20).31C4, D20.44(C2×C4), (C8×D5)⋊19C22, C23.39(C4×D5), C8.65(C22×D5), C4.66(C23×D5), C8⋊D5⋊21C22, C10.51(C23×C4), C5⋊2C8.31C23, (C4×D5).70C23, (C2×C20).880C23, C20.180(C22×C4), Dic10.47(C2×C4), (C2×Dic10).32C4, C4○D20.58C22, D10.21(C22×C4), (C22×C4).441D10, C4.Dic5⋊39C22, Dic5.20(C22×C4), (C22×C20).544C22, C5⋊4(C2×C8○D4), (D5×C2×C8)⋊25C2, C4.121(C2×C4×D5), C5⋊D4.9(C2×C4), (C2×C8⋊D5)⋊29C2, C22.11(C2×C4×D5), C2.31(D5×C22×C4), (C4×D5).59(C2×C4), (C2×C4).119(C4×D5), (C2×C5⋊D4).28C4, (C2×C20).409(C2×C4), (C2×C4○D20).29C2, (C2×C4.Dic5)⋊33C2, (C2×C4×D5).385C22, (C22×D5).82(C2×C4), (C2×C4).824(C22×D5), (C2×C10).257(C22×C4), (C22×C10).171(C2×C4), (C2×C5⋊2C8).335C22, (C2×Dic5).116(C2×C4), SmallGroup(320,1410)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D20.3C4
G = < a,b,c,d | a2=b20=c2=1, d4=b10, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, cd=dc >
Subgroups: 718 in 266 conjugacy classes, 151 normal (29 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C22×C8, C22×C8, C2×M4(2), C8○D4, C2×C4○D4, C5⋊2C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C2×C8○D4, C8×D5, C8⋊D5, C2×C5⋊2C8, C4.Dic5, C2×C40, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, D5×C2×C8, C2×C8⋊D5, D20.3C4, C2×C4.Dic5, C22×C40, C2×C4○D20, C2×D20.3C4
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, D10, C8○D4, C23×C4, C4×D5, C22×D5, C2×C8○D4, C2×C4×D5, C23×D5, D20.3C4, D5×C22×C4, C2×D20.3C4
►On 160 points
Generators in S160
(1 59)(2 60)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 58)(21 116)(22 117)(23 118)(24 119)(25 120)(26 101)(27 102)(28 103)(29 104)(30 105)(31 106)(32 107)(33 108)(34 109)(35 110)(36 111)(37 112)(38 113)(39 114)(40 115)(61 130)(62 131)(63 132)(64 133)(65 134)(66 135)(67 136)(68 137)(69 138)(70 139)(71 140)(72 121)(73 122)(74 123)(75 124)(76 125)(77 126)(78 127)(79 128)(80 129)(81 157)(82 158)(83 159)(84 160)(85 141)(86 142)(87 143)(88 144)(89 145)(90 146)(91 147)(92 148)(93 149)(94 150)(95 151)(96 152)(97 153)(98 154)(99 155)(100 156)
(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 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 60)(10 59)(11 58)(12 57)(13 56)(14 55)(15 54)(16 53)(17 52)(18 51)(19 50)(20 49)(21 105)(22 104)(23 103)(24 102)(25 101)(26 120)(27 119)(28 118)(29 117)(30 116)(31 115)(32 114)(33 113)(34 112)(35 111)(36 110)(37 109)(38 108)(39 107)(40 106)(61 135)(62 134)(63 133)(64 132)(65 131)(66 130)(67 129)(68 128)(69 127)(70 126)(71 125)(72 124)(73 123)(74 122)(75 121)(76 140)(77 139)(78 138)(79 137)(80 136)(81 154)(82 153)(83 152)(84 151)(85 150)(86 149)(87 148)(88 147)(89 146)(90 145)(91 144)(92 143)(93 142)(94 141)(95 160)(96 159)(97 158)(98 157)(99 156)(100 155)
(1 85 69 116 11 95 79 106)(2 86 70 117 12 96 80 107)(3 87 71 118 13 97 61 108)(4 88 72 119 14 98 62 109)(5 89 73 120 15 99 63 110)(6 90 74 101 16 100 64 111)(7 91 75 102 17 81 65 112)(8 92 76 103 18 82 66 113)(9 93 77 104 19 83 67 114)(10 94 78 105 20 84 68 115)(21 49 151 128 31 59 141 138)(22 50 152 129 32 60 142 139)(23 51 153 130 33 41 143 140)(24 52 154 131 34 42 144 121)(25 53 155 132 35 43 145 122)(26 54 156 133 36 44 146 123)(27 55 157 134 37 45 147 124)(28 56 158 135 38 46 148 125)(29 57 159 136 39 47 149 126)(30 58 160 137 40 48 150 127)
G:=sub<Sym(160)| (1,59)(2,60)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,116)(22,117)(23,118)(24,119)(25,120)(26,101)(27,102)(28,103)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,113)(39,114)(40,115)(61,130)(62,131)(63,132)(64,133)(65,134)(66,135)(67,136)(68,137)(69,138)(70,139)(71,140)(72,121)(73,122)(74,123)(75,124)(76,125)(77,126)(78,127)(79,128)(80,129)(81,157)(82,158)(83,159)(84,160)(85,141)(86,142)(87,143)(88,144)(89,145)(90,146)(91,147)(92,148)(93,149)(94,150)(95,151)(96,152)(97,153)(98,154)(99,155)(100,156), (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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,105)(22,104)(23,103)(24,102)(25,101)(26,120)(27,119)(28,118)(29,117)(30,116)(31,115)(32,114)(33,113)(34,112)(35,111)(36,110)(37,109)(38,108)(39,107)(40,106)(61,135)(62,134)(63,133)(64,132)(65,131)(66,130)(67,129)(68,128)(69,127)(70,126)(71,125)(72,124)(73,123)(74,122)(75,121)(76,140)(77,139)(78,138)(79,137)(80,136)(81,154)(82,153)(83,152)(84,151)(85,150)(86,149)(87,148)(88,147)(89,146)(90,145)(91,144)(92,143)(93,142)(94,141)(95,160)(96,159)(97,158)(98,157)(99,156)(100,155), (1,85,69,116,11,95,79,106)(2,86,70,117,12,96,80,107)(3,87,71,118,13,97,61,108)(4,88,72,119,14,98,62,109)(5,89,73,120,15,99,63,110)(6,90,74,101,16,100,64,111)(7,91,75,102,17,81,65,112)(8,92,76,103,18,82,66,113)(9,93,77,104,19,83,67,114)(10,94,78,105,20,84,68,115)(21,49,151,128,31,59,141,138)(22,50,152,129,32,60,142,139)(23,51,153,130,33,41,143,140)(24,52,154,131,34,42,144,121)(25,53,155,132,35,43,145,122)(26,54,156,133,36,44,146,123)(27,55,157,134,37,45,147,124)(28,56,158,135,38,46,148,125)(29,57,159,136,39,47,149,126)(30,58,160,137,40,48,150,127)>;
G:=Group( (1,59)(2,60)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,58)(21,116)(22,117)(23,118)(24,119)(25,120)(26,101)(27,102)(28,103)(29,104)(30,105)(31,106)(32,107)(33,108)(34,109)(35,110)(36,111)(37,112)(38,113)(39,114)(40,115)(61,130)(62,131)(63,132)(64,133)(65,134)(66,135)(67,136)(68,137)(69,138)(70,139)(71,140)(72,121)(73,122)(74,123)(75,124)(76,125)(77,126)(78,127)(79,128)(80,129)(81,157)(82,158)(83,159)(84,160)(85,141)(86,142)(87,143)(88,144)(89,145)(90,146)(91,147)(92,148)(93,149)(94,150)(95,151)(96,152)(97,153)(98,154)(99,155)(100,156), (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), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,42)(8,41)(9,60)(10,59)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,105)(22,104)(23,103)(24,102)(25,101)(26,120)(27,119)(28,118)(29,117)(30,116)(31,115)(32,114)(33,113)(34,112)(35,111)(36,110)(37,109)(38,108)(39,107)(40,106)(61,135)(62,134)(63,133)(64,132)(65,131)(66,130)(67,129)(68,128)(69,127)(70,126)(71,125)(72,124)(73,123)(74,122)(75,121)(76,140)(77,139)(78,138)(79,137)(80,136)(81,154)(82,153)(83,152)(84,151)(85,150)(86,149)(87,148)(88,147)(89,146)(90,145)(91,144)(92,143)(93,142)(94,141)(95,160)(96,159)(97,158)(98,157)(99,156)(100,155), (1,85,69,116,11,95,79,106)(2,86,70,117,12,96,80,107)(3,87,71,118,13,97,61,108)(4,88,72,119,14,98,62,109)(5,89,73,120,15,99,63,110)(6,90,74,101,16,100,64,111)(7,91,75,102,17,81,65,112)(8,92,76,103,18,82,66,113)(9,93,77,104,19,83,67,114)(10,94,78,105,20,84,68,115)(21,49,151,128,31,59,141,138)(22,50,152,129,32,60,142,139)(23,51,153,130,33,41,143,140)(24,52,154,131,34,42,144,121)(25,53,155,132,35,43,145,122)(26,54,156,133,36,44,146,123)(27,55,157,134,37,45,147,124)(28,56,158,135,38,46,148,125)(29,57,159,136,39,47,149,126)(30,58,160,137,40,48,150,127) );
G=PermutationGroup([[(1,59),(2,60),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,58),(21,116),(22,117),(23,118),(24,119),(25,120),(26,101),(27,102),(28,103),(29,104),(30,105),(31,106),(32,107),(33,108),(34,109),(35,110),(36,111),(37,112),(38,113),(39,114),(40,115),(61,130),(62,131),(63,132),(64,133),(65,134),(66,135),(67,136),(68,137),(69,138),(70,139),(71,140),(72,121),(73,122),(74,123),(75,124),(76,125),(77,126),(78,127),(79,128),(80,129),(81,157),(82,158),(83,159),(84,160),(85,141),(86,142),(87,143),(88,144),(89,145),(90,146),(91,147),(92,148),(93,149),(94,150),(95,151),(96,152),(97,153),(98,154),(99,155),(100,156)], [(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)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,42),(8,41),(9,60),(10,59),(11,58),(12,57),(13,56),(14,55),(15,54),(16,53),(17,52),(18,51),(19,50),(20,49),(21,105),(22,104),(23,103),(24,102),(25,101),(26,120),(27,119),(28,118),(29,117),(30,116),(31,115),(32,114),(33,113),(34,112),(35,111),(36,110),(37,109),(38,108),(39,107),(40,106),(61,135),(62,134),(63,133),(64,132),(65,131),(66,130),(67,129),(68,128),(69,127),(70,126),(71,125),(72,124),(73,123),(74,122),(75,121),(76,140),(77,139),(78,138),(79,137),(80,136),(81,154),(82,153),(83,152),(84,151),(85,150),(86,149),(87,148),(88,147),(89,146),(90,145),(91,144),(92,143),(93,142),(94,141),(95,160),(96,159),(97,158),(98,157),(99,156),(100,155)], [(1,85,69,116,11,95,79,106),(2,86,70,117,12,96,80,107),(3,87,71,118,13,97,61,108),(4,88,72,119,14,98,62,109),(5,89,73,120,15,99,63,110),(6,90,74,101,16,100,64,111),(7,91,75,102,17,81,65,112),(8,92,76,103,18,82,66,113),(9,93,77,104,19,83,67,114),(10,94,78,105,20,84,68,115),(21,49,151,128,31,59,141,138),(22,50,152,129,32,60,142,139),(23,51,153,130,33,41,143,140),(24,52,154,131,34,42,144,121),(25,53,155,132,35,43,145,122),(26,54,156,133,36,44,146,123),(27,55,157,134,37,45,147,124),(28,56,158,135,38,46,148,125),(29,57,159,136,39,47,149,126),(30,58,160,137,40,48,150,127)]])
104 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 8M | ··· | 8T | 10A | ··· | 10N | 20A | ··· | 20P | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D5 | D10 | D10 | C8○D4 | C4×D5 | C4×D5 | D20.3C4 |
| kernel | C2×D20.3C4 | D5×C2×C8 | C2×C8⋊D5 | D20.3C4 | C2×C4.Dic5 | C22×C40 | C2×C4○D20 | C2×Dic10 | C2×D20 | C4○D20 | C2×C5⋊D4 | C22×C8 | C2×C8 | C22×C4 | C10 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 2 | 2 | 8 | 4 | 2 | 12 | 2 | 8 | 12 | 4 | 32 |
Matrix representation of C2×D20.3C4 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 36 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 6 | 40 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 9 | 36 | 0 | 0 |
| 0 | 16 | 32 | 0 | 0 |
| 0 | 0 | 0 | 6 | 40 |
| 0 | 0 | 0 | 35 | 35 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 14 | 0 | 0 | 0 |
| 0 | 0 | 14 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 32 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,9,0,0,0,0,36,32,0,0,0,0,0,6,1,0,0,0,40,0],[1,0,0,0,0,0,9,16,0,0,0,36,32,0,0,0,0,0,6,35,0,0,0,40,35],[32,0,0,0,0,0,14,0,0,0,0,0,14,0,0,0,0,0,32,0,0,0,0,0,32] >;
C2×D20.3C4 in GAP, Magma, Sage, TeX
C_2\times D_{20}._3C_4 % in TeX
G:=Group("C2xD20.3C4"); // GroupNames label
G:=SmallGroup(320,1410);
// by ID
G=gap.SmallGroup(320,1410);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,80,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^2=1,d^4=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,c*d=d*c>;
// generators/relations