direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×D40, C20⋊5D8, C42.259D10, C5⋊2(C4×D8), (C4×C8)⋊7D5, C8⋊10(C4×D5), C40⋊32(C2×C4), (C4×C40)⋊12C2, (C4×D20)⋊1C2, C10.2(C2×D8), C2.1(C2×D40), D20⋊18(C2×C4), C40⋊5C4⋊28C2, C2.10(C4×D20), C10.37(C4×D4), (C2×C4).61D20, (C2×D40).14C2, C10.3(C4○D8), (C2×C8).286D10, (C2×C20).351D4, D20⋊5C4⋊43C2, C22.28(C2×D20), C20.217(C4○D4), C4.101(C4○D20), C2.2(D40⋊7C2), (C4×C20).326C22, (C2×C40).346C22, (C2×C20).721C23, C20.161(C22×C4), (C2×D20).194C22, C4⋊Dic5.263C22, C4.60(C2×C4×D5), (C2×C10).104(C2×D4), (C2×C4).664(C22×D5), SmallGroup(320,319)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×D40
G = < a,b,c | a4=b40=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 662 in 134 conjugacy classes, 55 normal (31 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, Dic5, C20, C20, C20, D10, C2×C10, C4×C8, D4⋊C4, C2.D8, C4×D4, C2×D8, C40, C40, C4×D5, D20, D20, C2×Dic5, C2×C20, C22×D5, C4×D8, D40, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×C4×D5, C2×D20, C40⋊5C4, D20⋊5C4, C4×C40, C4×D20, C2×D40, C4×D40
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, D8, C22×C4, C2×D4, C4○D4, D10, C4×D4, C2×D8, C4○D8, C4×D5, D20, C22×D5, C4×D8, D40, C2×C4×D5, C2×D20, C4○D20, C4×D20, C2×D40, D40⋊7C2, C4×D40
►On 160 points
Generators in S160
(1 101 136 55)(2 102 137 56)(3 103 138 57)(4 104 139 58)(5 105 140 59)(6 106 141 60)(7 107 142 61)(8 108 143 62)(9 109 144 63)(10 110 145 64)(11 111 146 65)(12 112 147 66)(13 113 148 67)(14 114 149 68)(15 115 150 69)(16 116 151 70)(17 117 152 71)(18 118 153 72)(19 119 154 73)(20 120 155 74)(21 81 156 75)(22 82 157 76)(23 83 158 77)(24 84 159 78)(25 85 160 79)(26 86 121 80)(27 87 122 41)(28 88 123 42)(29 89 124 43)(30 90 125 44)(31 91 126 45)(32 92 127 46)(33 93 128 47)(34 94 129 48)(35 95 130 49)(36 96 131 50)(37 97 132 51)(38 98 133 52)(39 99 134 53)(40 100 135 54)
(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)
(1 150)(2 149)(3 148)(4 147)(5 146)(6 145)(7 144)(8 143)(9 142)(10 141)(11 140)(12 139)(13 138)(14 137)(15 136)(16 135)(17 134)(18 133)(19 132)(20 131)(21 130)(22 129)(23 128)(24 127)(25 126)(26 125)(27 124)(28 123)(29 122)(30 121)(31 160)(32 159)(33 158)(34 157)(35 156)(36 155)(37 154)(38 153)(39 152)(40 151)(41 89)(42 88)(43 87)(44 86)(45 85)(46 84)(47 83)(48 82)(49 81)(50 120)(51 119)(52 118)(53 117)(54 116)(55 115)(56 114)(57 113)(58 112)(59 111)(60 110)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 100)(71 99)(72 98)(73 97)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)
G:=sub<Sym(160)| (1,101,136,55)(2,102,137,56)(3,103,138,57)(4,104,139,58)(5,105,140,59)(6,106,141,60)(7,107,142,61)(8,108,143,62)(9,109,144,63)(10,110,145,64)(11,111,146,65)(12,112,147,66)(13,113,148,67)(14,114,149,68)(15,115,150,69)(16,116,151,70)(17,117,152,71)(18,118,153,72)(19,119,154,73)(20,120,155,74)(21,81,156,75)(22,82,157,76)(23,83,158,77)(24,84,159,78)(25,85,160,79)(26,86,121,80)(27,87,122,41)(28,88,123,42)(29,89,124,43)(30,90,125,44)(31,91,126,45)(32,92,127,46)(33,93,128,47)(34,94,129,48)(35,95,130,49)(36,96,131,50)(37,97,132,51)(38,98,133,52)(39,99,134,53)(40,100,135,54), (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), (1,150)(2,149)(3,148)(4,147)(5,146)(6,145)(7,144)(8,143)(9,142)(10,141)(11,140)(12,139)(13,138)(14,137)(15,136)(16,135)(17,134)(18,133)(19,132)(20,131)(21,130)(22,129)(23,128)(24,127)(25,126)(26,125)(27,124)(28,123)(29,122)(30,121)(31,160)(32,159)(33,158)(34,157)(35,156)(36,155)(37,154)(38,153)(39,152)(40,151)(41,89)(42,88)(43,87)(44,86)(45,85)(46,84)(47,83)(48,82)(49,81)(50,120)(51,119)(52,118)(53,117)(54,116)(55,115)(56,114)(57,113)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)>;
G:=Group( (1,101,136,55)(2,102,137,56)(3,103,138,57)(4,104,139,58)(5,105,140,59)(6,106,141,60)(7,107,142,61)(8,108,143,62)(9,109,144,63)(10,110,145,64)(11,111,146,65)(12,112,147,66)(13,113,148,67)(14,114,149,68)(15,115,150,69)(16,116,151,70)(17,117,152,71)(18,118,153,72)(19,119,154,73)(20,120,155,74)(21,81,156,75)(22,82,157,76)(23,83,158,77)(24,84,159,78)(25,85,160,79)(26,86,121,80)(27,87,122,41)(28,88,123,42)(29,89,124,43)(30,90,125,44)(31,91,126,45)(32,92,127,46)(33,93,128,47)(34,94,129,48)(35,95,130,49)(36,96,131,50)(37,97,132,51)(38,98,133,52)(39,99,134,53)(40,100,135,54), (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), (1,150)(2,149)(3,148)(4,147)(5,146)(6,145)(7,144)(8,143)(9,142)(10,141)(11,140)(12,139)(13,138)(14,137)(15,136)(16,135)(17,134)(18,133)(19,132)(20,131)(21,130)(22,129)(23,128)(24,127)(25,126)(26,125)(27,124)(28,123)(29,122)(30,121)(31,160)(32,159)(33,158)(34,157)(35,156)(36,155)(37,154)(38,153)(39,152)(40,151)(41,89)(42,88)(43,87)(44,86)(45,85)(46,84)(47,83)(48,82)(49,81)(50,120)(51,119)(52,118)(53,117)(54,116)(55,115)(56,114)(57,113)(58,112)(59,111)(60,110)(61,109)(62,108)(63,107)(64,106)(65,105)(66,104)(67,103)(68,102)(69,101)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90) );
G=PermutationGroup([[(1,101,136,55),(2,102,137,56),(3,103,138,57),(4,104,139,58),(5,105,140,59),(6,106,141,60),(7,107,142,61),(8,108,143,62),(9,109,144,63),(10,110,145,64),(11,111,146,65),(12,112,147,66),(13,113,148,67),(14,114,149,68),(15,115,150,69),(16,116,151,70),(17,117,152,71),(18,118,153,72),(19,119,154,73),(20,120,155,74),(21,81,156,75),(22,82,157,76),(23,83,158,77),(24,84,159,78),(25,85,160,79),(26,86,121,80),(27,87,122,41),(28,88,123,42),(29,89,124,43),(30,90,125,44),(31,91,126,45),(32,92,127,46),(33,93,128,47),(34,94,129,48),(35,95,130,49),(36,96,131,50),(37,97,132,51),(38,98,133,52),(39,99,134,53),(40,100,135,54)], [(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)], [(1,150),(2,149),(3,148),(4,147),(5,146),(6,145),(7,144),(8,143),(9,142),(10,141),(11,140),(12,139),(13,138),(14,137),(15,136),(16,135),(17,134),(18,133),(19,132),(20,131),(21,130),(22,129),(23,128),(24,127),(25,126),(26,125),(27,124),(28,123),(29,122),(30,121),(31,160),(32,159),(33,158),(34,157),(35,156),(36,155),(37,154),(38,153),(39,152),(40,151),(41,89),(42,88),(43,87),(44,86),(45,85),(46,84),(47,83),(48,82),(49,81),(50,120),(51,119),(52,118),(53,117),(54,116),(55,115),(56,114),(57,113),(58,112),(59,111),(60,110),(61,109),(62,108),(63,107),(64,106),(65,105),(66,104),(67,103),(68,102),(69,101),(70,100),(71,99),(72,98),(73,97),(74,96),(75,95),(76,94),(77,93),(78,92),(79,91),(80,90)]])
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10F | 20A | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 20 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D5 | D8 | C4○D4 | D10 | D10 | C4○D8 | C4×D5 | D20 | D40 | C4○D20 | D40⋊7C2 |
| kernel | C4×D40 | C40⋊5C4 | D20⋊5C4 | C4×C40 | C4×D20 | C2×D40 | D40 | C2×C20 | C4×C8 | C20 | C20 | C42 | C2×C8 | C10 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 8 | 8 | 16 | 8 | 16 |
Matrix representation of C4×D40 ►in GL4(𝔽41) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 35 | 0 | 0 |
| 6 | 35 | 0 | 0 |
| 0 | 0 | 18 | 3 |
| 0 | 0 | 38 | 36 |
| 6 | 1 | 0 | 0 |
| 6 | 35 | 0 | 0 |
| 0 | 0 | 25 | 16 |
| 0 | 0 | 2 | 16 |
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[40,6,0,0,35,35,0,0,0,0,18,38,0,0,3,36],[6,6,0,0,1,35,0,0,0,0,25,2,0,0,16,16] >;
C4×D40 in GAP, Magma, Sage, TeX
C_4\times D_{40} % in TeX
G:=Group("C4xD40"); // GroupNames label
G:=SmallGroup(320,319);
// by ID
G=gap.SmallGroup(320,319);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,58,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^4=b^40=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations