direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20⋊D4, C24.40D10, C20⋊9(C2×D4), (C2×C20)⋊13D4, (C2×D4)⋊39D10, Dic5⋊2(C2×D4), (C22×D4)⋊9D5, C10⋊2(C4⋊1D4), (C2×Dic5)⋊14D4, (D4×C10)⋊44C22, (C2×D20)⋊56C22, (C22×D20)⋊19C2, C22.149(D4×D5), (C2×C20).544C23, (C2×C10).298C24, (C4×Dic5)⋊68C22, (C22×C4).380D10, C10.145(C22×D4), (C23×C10).78C22, (C23×D5).77C22, C22.311(C23×D5), C23.135(C22×D5), (C22×C10).232C23, (C22×C20).276C22, (C2×Dic5).295C23, (C22×D5).129C23, (C22×Dic5).255C22, (D4×C2×C10)⋊6C2, C4⋊1(C2×C5⋊D4), C5⋊3(C2×C4⋊1D4), C2.105(C2×D4×D5), (C2×C4×Dic5)⋊12C2, (C2×C4)⋊10(C5⋊D4), (C2×C10).581(C2×D4), (C2×C5⋊D4)⋊47C22, (C22×C5⋊D4)⋊16C2, C2.18(C22×C5⋊D4), (C2×C4).627(C22×D5), C22.111(C2×C5⋊D4), SmallGroup(320,1475)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C20⋊D4
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b9, dbd=b-1, dcd=c-1 >
Subgroups: 2014 in 498 conjugacy classes, 143 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C10, C42, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×C42, C4⋊1D4, C22×D4, C22×D4, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C2×C4⋊1D4, C4×Dic5, C2×D20, C2×D20, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, C23×D5, C23×C10, C2×C4×Dic5, C20⋊D4, C22×D20, C22×C5⋊D4, D4×C2×C10, C2×C20⋊D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C4⋊1D4, C22×D4, C5⋊D4, C22×D5, C2×C4⋊1D4, D4×D5, C2×C5⋊D4, C23×D5, C20⋊D4, C2×D4×D5, C22×C5⋊D4, C2×C20⋊D4
►On 160 points
Generators in S160
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 109)(22 110)(23 111)(24 112)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(33 101)(34 102)(35 103)(36 104)(37 105)(38 106)(39 107)(40 108)(61 146)(62 147)(63 148)(64 149)(65 150)(66 151)(67 152)(68 153)(69 154)(70 155)(71 156)(72 157)(73 158)(74 159)(75 160)(76 141)(77 142)(78 143)(79 144)(80 145)(81 136)(82 137)(83 138)(84 139)(85 140)(86 121)(87 122)(88 123)(89 124)(90 125)(91 126)(92 127)(93 128)(94 129)(95 130)(96 131)(97 132)(98 133)(99 134)(100 135)
(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)
(1 114 123 149)(2 103 124 158)(3 112 125 147)(4 101 126 156)(5 110 127 145)(6 119 128 154)(7 108 129 143)(8 117 130 152)(9 106 131 141)(10 115 132 150)(11 104 133 159)(12 113 134 148)(13 102 135 157)(14 111 136 146)(15 120 137 155)(16 109 138 144)(17 118 139 153)(18 107 140 142)(19 116 121 151)(20 105 122 160)(21 83 79 45)(22 92 80 54)(23 81 61 43)(24 90 62 52)(25 99 63 41)(26 88 64 50)(27 97 65 59)(28 86 66 48)(29 95 67 57)(30 84 68 46)(31 93 69 55)(32 82 70 44)(33 91 71 53)(34 100 72 42)(35 89 73 51)(36 98 74 60)(37 87 75 49)(38 96 76 58)(39 85 77 47)(40 94 78 56)
(1 133)(2 132)(3 131)(4 130)(5 129)(6 128)(7 127)(8 126)(9 125)(10 124)(11 123)(12 122)(13 121)(14 140)(15 139)(16 138)(17 137)(18 136)(19 135)(20 134)(22 40)(23 39)(24 38)(25 37)(26 36)(27 35)(28 34)(29 33)(30 32)(41 87)(42 86)(43 85)(44 84)(45 83)(46 82)(47 81)(48 100)(49 99)(50 98)(51 97)(52 96)(53 95)(54 94)(55 93)(56 92)(57 91)(58 90)(59 89)(60 88)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(78 80)(101 117)(102 116)(103 115)(104 114)(105 113)(106 112)(107 111)(108 110)(118 120)(141 147)(142 146)(143 145)(148 160)(149 159)(150 158)(151 157)(152 156)(153 155)
G:=sub<Sym(160)| (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(61,146)(62,147)(63,148)(64,149)(65,150)(66,151)(67,152)(68,153)(69,154)(70,155)(71,156)(72,157)(73,158)(74,159)(75,160)(76,141)(77,142)(78,143)(79,144)(80,145)(81,136)(82,137)(83,138)(84,139)(85,140)(86,121)(87,122)(88,123)(89,124)(90,125)(91,126)(92,127)(93,128)(94,129)(95,130)(96,131)(97,132)(98,133)(99,134)(100,135), (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), (1,114,123,149)(2,103,124,158)(3,112,125,147)(4,101,126,156)(5,110,127,145)(6,119,128,154)(7,108,129,143)(8,117,130,152)(9,106,131,141)(10,115,132,150)(11,104,133,159)(12,113,134,148)(13,102,135,157)(14,111,136,146)(15,120,137,155)(16,109,138,144)(17,118,139,153)(18,107,140,142)(19,116,121,151)(20,105,122,160)(21,83,79,45)(22,92,80,54)(23,81,61,43)(24,90,62,52)(25,99,63,41)(26,88,64,50)(27,97,65,59)(28,86,66,48)(29,95,67,57)(30,84,68,46)(31,93,69,55)(32,82,70,44)(33,91,71,53)(34,100,72,42)(35,89,73,51)(36,98,74,60)(37,87,75,49)(38,96,76,58)(39,85,77,47)(40,94,78,56), (1,133)(2,132)(3,131)(4,130)(5,129)(6,128)(7,127)(8,126)(9,125)(10,124)(11,123)(12,122)(13,121)(14,140)(15,139)(16,138)(17,137)(18,136)(19,135)(20,134)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,87)(42,86)(43,85)(44,84)(45,83)(46,82)(47,81)(48,100)(49,99)(50,98)(51,97)(52,96)(53,95)(54,94)(55,93)(56,92)(57,91)(58,90)(59,89)(60,88)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(78,80)(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(118,120)(141,147)(142,146)(143,145)(148,160)(149,159)(150,158)(151,157)(152,156)(153,155)>;
G:=Group( (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(61,146)(62,147)(63,148)(64,149)(65,150)(66,151)(67,152)(68,153)(69,154)(70,155)(71,156)(72,157)(73,158)(74,159)(75,160)(76,141)(77,142)(78,143)(79,144)(80,145)(81,136)(82,137)(83,138)(84,139)(85,140)(86,121)(87,122)(88,123)(89,124)(90,125)(91,126)(92,127)(93,128)(94,129)(95,130)(96,131)(97,132)(98,133)(99,134)(100,135), (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), (1,114,123,149)(2,103,124,158)(3,112,125,147)(4,101,126,156)(5,110,127,145)(6,119,128,154)(7,108,129,143)(8,117,130,152)(9,106,131,141)(10,115,132,150)(11,104,133,159)(12,113,134,148)(13,102,135,157)(14,111,136,146)(15,120,137,155)(16,109,138,144)(17,118,139,153)(18,107,140,142)(19,116,121,151)(20,105,122,160)(21,83,79,45)(22,92,80,54)(23,81,61,43)(24,90,62,52)(25,99,63,41)(26,88,64,50)(27,97,65,59)(28,86,66,48)(29,95,67,57)(30,84,68,46)(31,93,69,55)(32,82,70,44)(33,91,71,53)(34,100,72,42)(35,89,73,51)(36,98,74,60)(37,87,75,49)(38,96,76,58)(39,85,77,47)(40,94,78,56), (1,133)(2,132)(3,131)(4,130)(5,129)(6,128)(7,127)(8,126)(9,125)(10,124)(11,123)(12,122)(13,121)(14,140)(15,139)(16,138)(17,137)(18,136)(19,135)(20,134)(22,40)(23,39)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(30,32)(41,87)(42,86)(43,85)(44,84)(45,83)(46,82)(47,81)(48,100)(49,99)(50,98)(51,97)(52,96)(53,95)(54,94)(55,93)(56,92)(57,91)(58,90)(59,89)(60,88)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(78,80)(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,111)(108,110)(118,120)(141,147)(142,146)(143,145)(148,160)(149,159)(150,158)(151,157)(152,156)(153,155) );
G=PermutationGroup([[(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,57),(9,58),(10,59),(11,60),(12,41),(13,42),(14,43),(15,44),(16,45),(17,46),(18,47),(19,48),(20,49),(21,109),(22,110),(23,111),(24,112),(25,113),(26,114),(27,115),(28,116),(29,117),(30,118),(31,119),(32,120),(33,101),(34,102),(35,103),(36,104),(37,105),(38,106),(39,107),(40,108),(61,146),(62,147),(63,148),(64,149),(65,150),(66,151),(67,152),(68,153),(69,154),(70,155),(71,156),(72,157),(73,158),(74,159),(75,160),(76,141),(77,142),(78,143),(79,144),(80,145),(81,136),(82,137),(83,138),(84,139),(85,140),(86,121),(87,122),(88,123),(89,124),(90,125),(91,126),(92,127),(93,128),(94,129),(95,130),(96,131),(97,132),(98,133),(99,134),(100,135)], [(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)], [(1,114,123,149),(2,103,124,158),(3,112,125,147),(4,101,126,156),(5,110,127,145),(6,119,128,154),(7,108,129,143),(8,117,130,152),(9,106,131,141),(10,115,132,150),(11,104,133,159),(12,113,134,148),(13,102,135,157),(14,111,136,146),(15,120,137,155),(16,109,138,144),(17,118,139,153),(18,107,140,142),(19,116,121,151),(20,105,122,160),(21,83,79,45),(22,92,80,54),(23,81,61,43),(24,90,62,52),(25,99,63,41),(26,88,64,50),(27,97,65,59),(28,86,66,48),(29,95,67,57),(30,84,68,46),(31,93,69,55),(32,82,70,44),(33,91,71,53),(34,100,72,42),(35,89,73,51),(36,98,74,60),(37,87,75,49),(38,96,76,58),(39,85,77,47),(40,94,78,56)], [(1,133),(2,132),(3,131),(4,130),(5,129),(6,128),(7,127),(8,126),(9,125),(10,124),(11,123),(12,122),(13,121),(14,140),(15,139),(16,138),(17,137),(18,136),(19,135),(20,134),(22,40),(23,39),(24,38),(25,37),(26,36),(27,35),(28,34),(29,33),(30,32),(41,87),(42,86),(43,85),(44,84),(45,83),(46,82),(47,81),(48,100),(49,99),(50,98),(51,97),(52,96),(53,95),(54,94),(55,93),(56,92),(57,91),(58,90),(59,89),(60,88),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70),(78,80),(101,117),(102,116),(103,115),(104,114),(105,113),(106,112),(107,111),(108,110),(118,120),(141,147),(142,146),(143,145),(148,160),(149,159),(150,158),(151,157),(152,156),(153,155)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10AD | 20A | ··· | 20H |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | C5⋊D4 | D4×D5 |
| kernel | C2×C20⋊D4 | C2×C4×Dic5 | C20⋊D4 | C22×D20 | C22×C5⋊D4 | D4×C2×C10 | C2×Dic5 | C2×C20 | C22×D4 | C22×C4 | C2×D4 | C24 | C2×C4 | C22 |
| # reps | 1 | 1 | 8 | 1 | 4 | 1 | 8 | 4 | 2 | 2 | 8 | 4 | 16 | 8 |
Matrix representation of C2×C20⋊D4 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 5 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 2 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 25 | 32 | 0 | 0 | 0 | 0 |
| 24 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 17 | 0 | 0 |
| 0 | 0 | 15 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 39 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,5,0,0,0,0,1,35,0,0,0,0,0,0,40,40,0,0,0,0,2,1],[25,24,0,0,0,0,32,16,0,0,0,0,0,0,21,15,0,0,0,0,17,20,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[40,40,0,0,0,0,0,1,0,0,0,0,0,0,6,6,0,0,0,0,1,35,0,0,0,0,0,0,1,1,0,0,0,0,0,40] >;
C2×C20⋊D4 in GAP, Magma, Sage, TeX
C_2\times C_{20}\rtimes D_4 % in TeX
G:=Group("C2xC20:D4"); // GroupNames label
G:=SmallGroup(320,1475);
// by ID
G=gap.SmallGroup(320,1475);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,675,297,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^9,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations