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), (C22×C40)⋊11C2, (C2×C40)⋊44C22, C20.290(C2×D4), (C2×C20).391D4, (C2×C4).100D20, C4.52(C23×D5), (C22×D20)⋊11C2, (C2×D20)⋊48C22, C2.24(C22×D20), C10.22(C22×D4), 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
Subgroups: 1822 in 338 conjugacy classes, 127 normal (13 characteristic)
C1, C2, C2 [×6], C2 [×8], C4, C4 [×3], C22 [×7], C22 [×32], C5, C8 [×4], C2×C4 [×6], D4 [×20], C23, C23 [×20], D5 [×8], C10, C10 [×6], C2×C8 [×6], D8 [×16], C22×C4, C2×D4 [×18], C24 [×2], C20, C20 [×3], D10 [×32], C2×C10 [×7], C22×C8, C2×D8 [×12], C22×D4 [×2], C40 [×4], D20 [×8], D20 [×12], C2×C20 [×6], C22×D5 [×20], C22×C10, C22×D8, D40 [×16], C2×C40 [×6], C2×D20 [×12], C2×D20 [×6], C22×C20, C23×D5 [×2], C2×D40 [×12], C22×C40, C22×D20 [×2], C22×D40
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, D8 [×4], C2×D4 [×6], C24, D10 [×7], C2×D8 [×6], C22×D4, D20 [×4], C22×D5 [×7], C22×D8, D40 [×4], C2×D20 [×6], C23×D5, C2×D40 [×6], C22×D20, C22×D40
Generators and relations
G = < a,b,c,d | a2=b2=c40=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►On 160 points
Generators in S160
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 100)(20 101)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 109)(29 110)(30 111)(31 112)(32 113)(33 114)(34 115)(35 116)(36 117)(37 118)(38 119)(39 120)(40 81)(41 133)(42 134)(43 135)(44 136)(45 137)(46 138)(47 139)(48 140)(49 141)(50 142)(51 143)(52 144)(53 145)(54 146)(55 147)(56 148)(57 149)(58 150)(59 151)(60 152)(61 153)(62 154)(63 155)(64 156)(65 157)(66 158)(67 159)(68 160)(69 121)(70 122)(71 123)(72 124)(73 125)(74 126)(75 127)(76 128)(77 129)(78 130)(79 131)(80 132)
(1 78)(2 79)(3 80)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 65)(29 66)(30 67)(31 68)(32 69)(33 70)(34 71)(35 72)(36 73)(37 74)(38 75)(39 76)(40 77)(81 129)(82 130)(83 131)(84 132)(85 133)(86 134)(87 135)(88 136)(89 137)(90 138)(91 139)(92 140)(93 141)(94 142)(95 143)(96 144)(97 145)(98 146)(99 147)(100 148)(101 149)(102 150)(103 151)(104 152)(105 153)(106 154)(107 155)(108 156)(109 157)(110 158)(111 159)(112 160)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 127)(120 128)
(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 62)(2 61)(3 60)(4 59)(5 58)(6 57)(7 56)(8 55)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 80)(24 79)(25 78)(26 77)(27 76)(28 75)(29 74)(30 73)(31 72)(32 71)(33 70)(34 69)(35 68)(36 67)(37 66)(38 65)(39 64)(40 63)(81 155)(82 154)(83 153)(84 152)(85 151)(86 150)(87 149)(88 148)(89 147)(90 146)(91 145)(92 144)(93 143)(94 142)(95 141)(96 140)(97 139)(98 138)(99 137)(100 136)(101 135)(102 134)(103 133)(104 132)(105 131)(106 130)(107 129)(108 128)(109 127)(110 126)(111 125)(112 124)(113 123)(114 122)(115 121)(116 160)(117 159)(118 158)(119 157)(120 156)
G:=sub<Sym(160)| (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,109)(29,110)(30,111)(31,112)(32,113)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,81)(41,133)(42,134)(43,135)(44,136)(45,137)(46,138)(47,139)(48,140)(49,141)(50,142)(51,143)(52,144)(53,145)(54,146)(55,147)(56,148)(57,149)(58,150)(59,151)(60,152)(61,153)(62,154)(63,155)(64,156)(65,157)(66,158)(67,159)(68,160)(69,121)(70,122)(71,123)(72,124)(73,125)(74,126)(75,127)(76,128)(77,129)(78,130)(79,131)(80,132), (1,78)(2,79)(3,80)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74)(38,75)(39,76)(40,77)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,141)(94,142)(95,143)(96,144)(97,145)(98,146)(99,147)(100,148)(101,149)(102,150)(103,151)(104,152)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128), (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), (1,62)(2,61)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,80)(24,79)(25,78)(26,77)(27,76)(28,75)(29,74)(30,73)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(81,155)(82,154)(83,153)(84,152)(85,151)(86,150)(87,149)(88,148)(89,147)(90,146)(91,145)(92,144)(93,143)(94,142)(95,141)(96,140)(97,139)(98,138)(99,137)(100,136)(101,135)(102,134)(103,133)(104,132)(105,131)(106,130)(107,129)(108,128)(109,127)(110,126)(111,125)(112,124)(113,123)(114,122)(115,121)(116,160)(117,159)(118,158)(119,157)(120,156)>;
G:=Group( (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,109)(29,110)(30,111)(31,112)(32,113)(33,114)(34,115)(35,116)(36,117)(37,118)(38,119)(39,120)(40,81)(41,133)(42,134)(43,135)(44,136)(45,137)(46,138)(47,139)(48,140)(49,141)(50,142)(51,143)(52,144)(53,145)(54,146)(55,147)(56,148)(57,149)(58,150)(59,151)(60,152)(61,153)(62,154)(63,155)(64,156)(65,157)(66,158)(67,159)(68,160)(69,121)(70,122)(71,123)(72,124)(73,125)(74,126)(75,127)(76,128)(77,129)(78,130)(79,131)(80,132), (1,78)(2,79)(3,80)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,65)(29,66)(30,67)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74)(38,75)(39,76)(40,77)(81,129)(82,130)(83,131)(84,132)(85,133)(86,134)(87,135)(88,136)(89,137)(90,138)(91,139)(92,140)(93,141)(94,142)(95,143)(96,144)(97,145)(98,146)(99,147)(100,148)(101,149)(102,150)(103,151)(104,152)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128), (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), (1,62)(2,61)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,80)(24,79)(25,78)(26,77)(27,76)(28,75)(29,74)(30,73)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(81,155)(82,154)(83,153)(84,152)(85,151)(86,150)(87,149)(88,148)(89,147)(90,146)(91,145)(92,144)(93,143)(94,142)(95,141)(96,140)(97,139)(98,138)(99,137)(100,136)(101,135)(102,134)(103,133)(104,132)(105,131)(106,130)(107,129)(108,128)(109,127)(110,126)(111,125)(112,124)(113,123)(114,122)(115,121)(116,160)(117,159)(118,158)(119,157)(120,156) );
G=PermutationGroup([(1,82),(2,83),(3,84),(4,85),(5,86),(6,87),(7,88),(8,89),(9,90),(10,91),(11,92),(12,93),(13,94),(14,95),(15,96),(16,97),(17,98),(18,99),(19,100),(20,101),(21,102),(22,103),(23,104),(24,105),(25,106),(26,107),(27,108),(28,109),(29,110),(30,111),(31,112),(32,113),(33,114),(34,115),(35,116),(36,117),(37,118),(38,119),(39,120),(40,81),(41,133),(42,134),(43,135),(44,136),(45,137),(46,138),(47,139),(48,140),(49,141),(50,142),(51,143),(52,144),(53,145),(54,146),(55,147),(56,148),(57,149),(58,150),(59,151),(60,152),(61,153),(62,154),(63,155),(64,156),(65,157),(66,158),(67,159),(68,160),(69,121),(70,122),(71,123),(72,124),(73,125),(74,126),(75,127),(76,128),(77,129),(78,130),(79,131),(80,132)], [(1,78),(2,79),(3,80),(4,41),(5,42),(6,43),(7,44),(8,45),(9,46),(10,47),(11,48),(12,49),(13,50),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,61),(25,62),(26,63),(27,64),(28,65),(29,66),(30,67),(31,68),(32,69),(33,70),(34,71),(35,72),(36,73),(37,74),(38,75),(39,76),(40,77),(81,129),(82,130),(83,131),(84,132),(85,133),(86,134),(87,135),(88,136),(89,137),(90,138),(91,139),(92,140),(93,141),(94,142),(95,143),(96,144),(97,145),(98,146),(99,147),(100,148),(101,149),(102,150),(103,151),(104,152),(105,153),(106,154),(107,155),(108,156),(109,157),(110,158),(111,159),(112,160),(113,121),(114,122),(115,123),(116,124),(117,125),(118,126),(119,127),(120,128)], [(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)], [(1,62),(2,61),(3,60),(4,59),(5,58),(6,57),(7,56),(8,55),(9,54),(10,53),(11,52),(12,51),(13,50),(14,49),(15,48),(16,47),(17,46),(18,45),(19,44),(20,43),(21,42),(22,41),(23,80),(24,79),(25,78),(26,77),(27,76),(28,75),(29,74),(30,73),(31,72),(32,71),(33,70),(34,69),(35,68),(36,67),(37,66),(38,65),(39,64),(40,63),(81,155),(82,154),(83,153),(84,152),(85,151),(86,150),(87,149),(88,148),(89,147),(90,146),(91,145),(92,144),(93,143),(94,142),(95,141),(96,140),(97,139),(98,138),(99,137),(100,136),(101,135),(102,134),(103,133),(104,132),(105,131),(106,130),(107,129),(108,128),(109,127),(110,126),(111,125),(112,124),(113,123),(114,122),(115,121),(116,160),(117,159),(118,158),(119,157),(120,156)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀