direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×D40, C40⋊9C23, D20⋊4C23, C20.55C24, C23.61D20, (C2×C10)⋊6D8, C10⋊1(C2×D8), C5⋊1(C22×D8), (C2×C8)⋊33D10, (C22×C8)⋊7D5, C8⋊8(C22×D5), C4.45(C2×D20), (C2×C40)⋊44C22, (C22×C40)⋊11C2, (C2×C4).100D20, (C2×C20).391D4, C20.290(C2×D4), C4.52(C23×D5), (C22×D20)⋊11C2, (C2×D20)⋊48C22, C10.22(C22×D4), C2.24(C22×D20), C22.70(C2×D20), (C2×C20).787C23, (C22×C4).443D10, (C22×C10).145D4, (C22×C20).526C22, (C2×C10).178(C2×D4), (C2×C4).736(C22×D5), SmallGroup(320,1412)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22×D40
G = < a,b,c,d | a2=b2=c40=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1822 in 338 conjugacy classes, 127 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, D4, C23, C23, D5, C10, C10, C2×C8, D8, C22×C4, C2×D4, C24, C20, C20, D10, C2×C10, C22×C8, C2×D8, C22×D4, C40, D20, D20, C2×C20, C22×D5, C22×C10, C22×D8, D40, C2×C40, C2×D20, C2×D20, C22×C20, C23×D5, C2×D40, C22×C40, C22×D20, C22×D40
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C24, D10, C2×D8, C22×D4, D20, C22×D5, C22×D8, D40, C2×D20, C23×D5, C2×D40, C22×D20, C22×D40
►On 160 points
Generators in S160
(1 86)(2 87)(3 88)(4 89)(5 90)(6 91)(7 92)(8 93)(9 94)(10 95)(11 96)(12 97)(13 98)(14 99)(15 100)(16 101)(17 102)(18 103)(19 104)(20 105)(21 106)(22 107)(23 108)(24 109)(25 110)(26 111)(27 112)(28 113)(29 114)(30 115)(31 116)(32 117)(33 118)(34 119)(35 120)(36 81)(37 82)(38 83)(39 84)(40 85)(41 123)(42 124)(43 125)(44 126)(45 127)(46 128)(47 129)(48 130)(49 131)(50 132)(51 133)(52 134)(53 135)(54 136)(55 137)(56 138)(57 139)(58 140)(59 141)(60 142)(61 143)(62 144)(63 145)(64 146)(65 147)(66 148)(67 149)(68 150)(69 151)(70 152)(71 153)(72 154)(73 155)(74 156)(75 157)(76 158)(77 159)(78 160)(79 121)(80 122)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 41)(38 42)(39 43)(40 44)(81 122)(82 123)(83 124)(84 125)(85 126)(86 127)(87 128)(88 129)(89 130)(90 131)(91 132)(92 133)(93 134)(94 135)(95 136)(96 137)(97 138)(98 139)(99 140)(100 141)(101 142)(102 143)(103 144)(104 145)(105 146)(106 147)(107 148)(108 149)(109 150)(110 151)(111 152)(112 153)(113 154)(114 155)(115 156)(116 157)(117 158)(118 159)(119 160)(120 121)
(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 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 60)(11 59)(12 58)(13 57)(14 56)(15 55)(16 54)(17 53)(18 52)(19 51)(20 50)(21 49)(22 48)(23 47)(24 46)(25 45)(26 44)(27 43)(28 42)(29 41)(30 80)(31 79)(32 78)(33 77)(34 76)(35 75)(36 74)(37 73)(38 72)(39 71)(40 70)(81 156)(82 155)(83 154)(84 153)(85 152)(86 151)(87 150)(88 149)(89 148)(90 147)(91 146)(92 145)(93 144)(94 143)(95 142)(96 141)(97 140)(98 139)(99 138)(100 137)(101 136)(102 135)(103 134)(104 133)(105 132)(106 131)(107 130)(108 129)(109 128)(110 127)(111 126)(112 125)(113 124)(114 123)(115 122)(116 121)(117 160)(118 159)(119 158)(120 157)
G:=sub<Sym(160)| (1,86)(2,87)(3,88)(4,89)(5,90)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,97)(13,98)(14,99)(15,100)(16,101)(17,102)(18,103)(19,104)(20,105)(21,106)(22,107)(23,108)(24,109)(25,110)(26,111)(27,112)(28,113)(29,114)(30,115)(31,116)(32,117)(33,118)(34,119)(35,120)(36,81)(37,82)(38,83)(39,84)(40,85)(41,123)(42,124)(43,125)(44,126)(45,127)(46,128)(47,129)(48,130)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,137)(56,138)(57,139)(58,140)(59,141)(60,142)(61,143)(62,144)(63,145)(64,146)(65,147)(66,148)(67,149)(68,150)(69,151)(70,152)(71,153)(72,154)(73,155)(74,156)(75,157)(76,158)(77,159)(78,160)(79,121)(80,122), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,41)(38,42)(39,43)(40,44)(81,122)(82,123)(83,124)(84,125)(85,126)(86,127)(87,128)(88,129)(89,130)(90,131)(91,132)(92,133)(93,134)(94,135)(95,136)(96,137)(97,138)(98,139)(99,140)(100,141)(101,142)(102,143)(103,144)(104,145)(105,146)(106,147)(107,148)(108,149)(109,150)(110,151)(111,152)(112,153)(113,154)(114,155)(115,156)(116,157)(117,158)(118,159)(119,160)(120,121), (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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,60)(11,59)(12,58)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(29,41)(30,80)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,70)(81,156)(82,155)(83,154)(84,153)(85,152)(86,151)(87,150)(88,149)(89,148)(90,147)(91,146)(92,145)(93,144)(94,143)(95,142)(96,141)(97,140)(98,139)(99,138)(100,137)(101,136)(102,135)(103,134)(104,133)(105,132)(106,131)(107,130)(108,129)(109,128)(110,127)(111,126)(112,125)(113,124)(114,123)(115,122)(116,121)(117,160)(118,159)(119,158)(120,157)>;
G:=Group( (1,86)(2,87)(3,88)(4,89)(5,90)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,97)(13,98)(14,99)(15,100)(16,101)(17,102)(18,103)(19,104)(20,105)(21,106)(22,107)(23,108)(24,109)(25,110)(26,111)(27,112)(28,113)(29,114)(30,115)(31,116)(32,117)(33,118)(34,119)(35,120)(36,81)(37,82)(38,83)(39,84)(40,85)(41,123)(42,124)(43,125)(44,126)(45,127)(46,128)(47,129)(48,130)(49,131)(50,132)(51,133)(52,134)(53,135)(54,136)(55,137)(56,138)(57,139)(58,140)(59,141)(60,142)(61,143)(62,144)(63,145)(64,146)(65,147)(66,148)(67,149)(68,150)(69,151)(70,152)(71,153)(72,154)(73,155)(74,156)(75,157)(76,158)(77,159)(78,160)(79,121)(80,122), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,41)(38,42)(39,43)(40,44)(81,122)(82,123)(83,124)(84,125)(85,126)(86,127)(87,128)(88,129)(89,130)(90,131)(91,132)(92,133)(93,134)(94,135)(95,136)(96,137)(97,138)(98,139)(99,140)(100,141)(101,142)(102,143)(103,144)(104,145)(105,146)(106,147)(107,148)(108,149)(109,150)(110,151)(111,152)(112,153)(113,154)(114,155)(115,156)(116,157)(117,158)(118,159)(119,160)(120,121), (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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,60)(11,59)(12,58)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,48)(23,47)(24,46)(25,45)(26,44)(27,43)(28,42)(29,41)(30,80)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,70)(81,156)(82,155)(83,154)(84,153)(85,152)(86,151)(87,150)(88,149)(89,148)(90,147)(91,146)(92,145)(93,144)(94,143)(95,142)(96,141)(97,140)(98,139)(99,138)(100,137)(101,136)(102,135)(103,134)(104,133)(105,132)(106,131)(107,130)(108,129)(109,128)(110,127)(111,126)(112,125)(113,124)(114,123)(115,122)(116,121)(117,160)(118,159)(119,158)(120,157) );
G=PermutationGroup([[(1,86),(2,87),(3,88),(4,89),(5,90),(6,91),(7,92),(8,93),(9,94),(10,95),(11,96),(12,97),(13,98),(14,99),(15,100),(16,101),(17,102),(18,103),(19,104),(20,105),(21,106),(22,107),(23,108),(24,109),(25,110),(26,111),(27,112),(28,113),(29,114),(30,115),(31,116),(32,117),(33,118),(34,119),(35,120),(36,81),(37,82),(38,83),(39,84),(40,85),(41,123),(42,124),(43,125),(44,126),(45,127),(46,128),(47,129),(48,130),(49,131),(50,132),(51,133),(52,134),(53,135),(54,136),(55,137),(56,138),(57,139),(58,140),(59,141),(60,142),(61,143),(62,144),(63,145),(64,146),(65,147),(66,148),(67,149),(68,150),(69,151),(70,152),(71,153),(72,154),(73,155),(74,156),(75,157),(76,158),(77,159),(78,160),(79,121),(80,122)], [(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,61),(18,62),(19,63),(20,64),(21,65),(22,66),(23,67),(24,68),(25,69),(26,70),(27,71),(28,72),(29,73),(30,74),(31,75),(32,76),(33,77),(34,78),(35,79),(36,80),(37,41),(38,42),(39,43),(40,44),(81,122),(82,123),(83,124),(84,125),(85,126),(86,127),(87,128),(88,129),(89,130),(90,131),(91,132),(92,133),(93,134),(94,135),(95,136),(96,137),(97,138),(98,139),(99,140),(100,141),(101,142),(102,143),(103,144),(104,145),(105,146),(106,147),(107,148),(108,149),(109,150),(110,151),(111,152),(112,153),(113,154),(114,155),(115,156),(116,157),(117,158),(118,159),(119,160),(120,121)], [(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)], [(1,69),(2,68),(3,67),(4,66),(5,65),(6,64),(7,63),(8,62),(9,61),(10,60),(11,59),(12,58),(13,57),(14,56),(15,55),(16,54),(17,53),(18,52),(19,51),(20,50),(21,49),(22,48),(23,47),(24,46),(25,45),(26,44),(27,43),(28,42),(29,41),(30,80),(31,79),(32,78),(33,77),(34,76),(35,75),(36,74),(37,73),(38,72),(39,71),(40,70),(81,156),(82,155),(83,154),(84,153),(85,152),(86,151),(87,150),(88,149),(89,148),(90,147),(91,146),(92,145),(93,144),(94,143),(95,142),(96,141),(97,140),(98,139),(99,138),(100,137),(101,136),(102,135),(103,134),(104,133),(105,132),(106,131),(107,130),(108,129),(109,128),(110,127),(111,126),(112,125),(113,124),(114,123),(115,122),(116,121),(117,160),(118,159),(119,158),(120,157)]])
92 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | 4B | 4C | 4D | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10N | 20A | ··· | 20P | 40A | ··· | 40AF |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | ··· | 1 | 20 | ··· | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 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 | D8 | D10 | D10 | D20 | D20 | D40 |
| kernel | C22×D40 | C2×D40 | C22×C40 | C22×D20 | C2×C20 | C22×C10 | C22×C8 | C2×C10 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 12 | 1 | 2 | 3 | 1 | 2 | 8 | 12 | 2 | 12 | 4 | 32 |
Matrix representation of C22×D40 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 39 | 3 |
| 0 | 0 | 17 | 15 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 7 | 1 |
| 0 | 0 | 34 | 34 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,1,0,0,0,0,39,17,0,0,3,15],[1,0,0,0,0,40,0,0,0,0,7,34,0,0,1,34] >;
C22×D40 in GAP, Magma, Sage, TeX
C_2^2\times D_{40} % in TeX
G:=Group("C2^2xD40"); // GroupNames label
G:=SmallGroup(320,1412);
// by ID
G=gap.SmallGroup(320,1412);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,675,192,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^40=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