direct product, abelian, monomial, 2-elementary
Aliases: C22×C40, SmallGroup(160,190)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C22×C40 |
| C1 — C22×C40 |
| C1 — C22×C40 |
Generators and relations for C22×C40
G = < a,b,c | a2=b2=c40=1, ab=ba, ac=ca, bc=cb >
Subgroups: 76, all normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C2×C4, C23, C10, C10, C2×C8, C22×C4, C20, C20, C2×C10, C22×C8, C40, C2×C20, C22×C10, C2×C40, C22×C20, C22×C40
Quotients: C1, C2, C4, C22, C5, C8, C2×C4, C23, C10, C2×C8, C22×C4, C20, C2×C10, C22×C8, C40, C2×C20, C22×C10, C2×C40, C22×C20, C22×C40
►Regular action on 160 points
Generators in S160
(1 88)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 98)(12 99)(13 100)(14 101)(15 102)(16 103)(17 104)(18 105)(19 106)(20 107)(21 108)(22 109)(23 110)(24 111)(25 112)(26 113)(27 114)(28 115)(29 116)(30 117)(31 118)(32 119)(33 120)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 87)(41 135)(42 136)(43 137)(44 138)(45 139)(46 140)(47 141)(48 142)(49 143)(50 144)(51 145)(52 146)(53 147)(54 148)(55 149)(56 150)(57 151)(58 152)(59 153)(60 154)(61 155)(62 156)(63 157)(64 158)(65 159)(66 160)(67 121)(68 122)(69 123)(70 124)(71 125)(72 126)(73 127)(74 128)(75 129)(76 130)(77 131)(78 132)(79 133)(80 134)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 79)(22 80)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 57)(40 58)(81 146)(82 147)(83 148)(84 149)(85 150)(86 151)(87 152)(88 153)(89 154)(90 155)(91 156)(92 157)(93 158)(94 159)(95 160)(96 121)(97 122)(98 123)(99 124)(100 125)(101 126)(102 127)(103 128)(104 129)(105 130)(106 131)(107 132)(108 133)(109 134)(110 135)(111 136)(112 137)(113 138)(114 139)(115 140)(116 141)(117 142)(118 143)(119 144)(120 145)
(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)
G:=sub<Sym(160)| (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,104)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,112)(26,113)(27,114)(28,115)(29,116)(30,117)(31,118)(32,119)(33,120)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,87)(41,135)(42,136)(43,137)(44,138)(45,139)(46,140)(47,141)(48,142)(49,143)(50,144)(51,145)(52,146)(53,147)(54,148)(55,149)(56,150)(57,151)(58,152)(59,153)(60,154)(61,155)(62,156)(63,157)(64,158)(65,159)(66,160)(67,121)(68,122)(69,123)(70,124)(71,125)(72,126)(73,127)(74,128)(75,129)(76,130)(77,131)(78,132)(79,133)(80,134), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,79)(22,80)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,57)(40,58)(81,146)(82,147)(83,148)(84,149)(85,150)(86,151)(87,152)(88,153)(89,154)(90,155)(91,156)(92,157)(93,158)(94,159)(95,160)(96,121)(97,122)(98,123)(99,124)(100,125)(101,126)(102,127)(103,128)(104,129)(105,130)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)(117,142)(118,143)(119,144)(120,145), (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)>;
G:=Group( (1,88)(2,89)(3,90)(4,91)(5,92)(6,93)(7,94)(8,95)(9,96)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,104)(18,105)(19,106)(20,107)(21,108)(22,109)(23,110)(24,111)(25,112)(26,113)(27,114)(28,115)(29,116)(30,117)(31,118)(32,119)(33,120)(34,81)(35,82)(36,83)(37,84)(38,85)(39,86)(40,87)(41,135)(42,136)(43,137)(44,138)(45,139)(46,140)(47,141)(48,142)(49,143)(50,144)(51,145)(52,146)(53,147)(54,148)(55,149)(56,150)(57,151)(58,152)(59,153)(60,154)(61,155)(62,156)(63,157)(64,158)(65,159)(66,160)(67,121)(68,122)(69,123)(70,124)(71,125)(72,126)(73,127)(74,128)(75,129)(76,130)(77,131)(78,132)(79,133)(80,134), (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,77)(20,78)(21,79)(22,80)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,57)(40,58)(81,146)(82,147)(83,148)(84,149)(85,150)(86,151)(87,152)(88,153)(89,154)(90,155)(91,156)(92,157)(93,158)(94,159)(95,160)(96,121)(97,122)(98,123)(99,124)(100,125)(101,126)(102,127)(103,128)(104,129)(105,130)(106,131)(107,132)(108,133)(109,134)(110,135)(111,136)(112,137)(113,138)(114,139)(115,140)(116,141)(117,142)(118,143)(119,144)(120,145), (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) );
G=PermutationGroup([[(1,88),(2,89),(3,90),(4,91),(5,92),(6,93),(7,94),(8,95),(9,96),(10,97),(11,98),(12,99),(13,100),(14,101),(15,102),(16,103),(17,104),(18,105),(19,106),(20,107),(21,108),(22,109),(23,110),(24,111),(25,112),(26,113),(27,114),(28,115),(29,116),(30,117),(31,118),(32,119),(33,120),(34,81),(35,82),(36,83),(37,84),(38,85),(39,86),(40,87),(41,135),(42,136),(43,137),(44,138),(45,139),(46,140),(47,141),(48,142),(49,143),(50,144),(51,145),(52,146),(53,147),(54,148),(55,149),(56,150),(57,151),(58,152),(59,153),(60,154),(61,155),(62,156),(63,157),(64,158),(65,159),(66,160),(67,121),(68,122),(69,123),(70,124),(71,125),(72,126),(73,127),(74,128),(75,129),(76,130),(77,131),(78,132),(79,133),(80,134)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,73),(16,74),(17,75),(18,76),(19,77),(20,78),(21,79),(22,80),(23,41),(24,42),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,49),(32,50),(33,51),(34,52),(35,53),(36,54),(37,55),(38,56),(39,57),(40,58),(81,146),(82,147),(83,148),(84,149),(85,150),(86,151),(87,152),(88,153),(89,154),(90,155),(91,156),(92,157),(93,158),(94,159),(95,160),(96,121),(97,122),(98,123),(99,124),(100,125),(101,126),(102,127),(103,128),(104,129),(105,130),(106,131),(107,132),(108,133),(109,134),(110,135),(111,136),(112,137),(113,138),(114,139),(115,140),(116,141),(117,142),(118,143),(119,144),(120,145)], [(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)]])
C22×C40 is a maximal subgroup of
C40.91D4 (C2×C40)⋊15C4 C20.39C42 C20.40C42 C20.42C42 C20.65(C4⋊C4) C23.22D20 (C22×C8)⋊D5 C40⋊32D4 C23.23D20 C40⋊30D4 C40⋊29D4 C40.82D4
160 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 5A | 5B | 5C | 5D | 8A | ··· | 8P | 10A | ··· | 10AB | 20A | ··· | 20AF | 40A | ··· | 40BL |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
160 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C8 | C10 | C10 | C20 | C20 | C40 |
| kernel | C22×C40 | C2×C40 | C22×C20 | C2×C20 | C22×C10 | C22×C8 | C2×C10 | C2×C8 | C22×C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 6 | 1 | 6 | 2 | 4 | 16 | 24 | 4 | 24 | 8 | 64 |
Matrix representation of C22×C40 ►in GL3(𝔽41) generated by
| 40 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 40 |
| 1 | 0 | 0 |
| 0 | 40 | 0 |
| 0 | 0 | 1 |
| 19 | 0 | 0 |
| 0 | 22 | 0 |
| 0 | 0 | 18 |
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,40],[1,0,0,0,40,0,0,0,1],[19,0,0,0,22,0,0,0,18] >;
C22×C40 in GAP, Magma, Sage, TeX
C_2^2\times C_{40} % in TeX
G:=Group("C2^2xC40"); // GroupNames label
G:=SmallGroup(160,190);
// by ID
G=gap.SmallGroup(160,190);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-2,240,88]);
// Polycyclic
G:=Group<a,b,c|a^2=b^2=c^40=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations