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 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×8], C5, C8 [×2], C8, C2×C4 [×3], C2×C4 [×6], D4 [×6], C23 [×2], D5 [×4], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], D8 [×4], C22×C4 [×2], C2×D4 [×2], Dic5 [×2], C20 [×2], C20 [×2], C20, D10 [×8], C2×C10, C4×C8, D4⋊C4 [×2], C2.D8, C4×D4 [×2], C2×D8, C40 [×2], C40, C4×D5 [×4], D20 [×4], D20 [×2], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], C4×D8, D40 [×4], C4⋊Dic5 [×2], D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×C4×D5 [×2], C2×D20 [×2], C40⋊5C4, D20⋊5C4 [×2], C4×C40, C4×D20 [×2], C2×D40, C4×D40
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, D8 [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×D8, C4○D8, C4×D5 [×2], D20 [×2], C22×D5, C4×D8, D40 [×2], C2×C4×D5, C2×D20, C4○D20, C4×D20, C2×D40, D40⋊7C2, C4×D40
►On 160 points
Generators in S160
(1 58 97 155)(2 59 98 156)(3 60 99 157)(4 61 100 158)(5 62 101 159)(6 63 102 160)(7 64 103 121)(8 65 104 122)(9 66 105 123)(10 67 106 124)(11 68 107 125)(12 69 108 126)(13 70 109 127)(14 71 110 128)(15 72 111 129)(16 73 112 130)(17 74 113 131)(18 75 114 132)(19 76 115 133)(20 77 116 134)(21 78 117 135)(22 79 118 136)(23 80 119 137)(24 41 120 138)(25 42 81 139)(26 43 82 140)(27 44 83 141)(28 45 84 142)(29 46 85 143)(30 47 86 144)(31 48 87 145)(32 49 88 146)(33 50 89 147)(34 51 90 148)(35 52 91 149)(36 53 92 150)(37 54 93 151)(38 55 94 152)(39 56 95 153)(40 57 96 154)
(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)
(1 111)(2 110)(3 109)(4 108)(5 107)(6 106)(7 105)(8 104)(9 103)(10 102)(11 101)(12 100)(13 99)(14 98)(15 97)(16 96)(17 95)(18 94)(19 93)(20 92)(21 91)(22 90)(23 89)(24 88)(25 87)(26 86)(27 85)(28 84)(29 83)(30 82)(31 81)(32 120)(33 119)(34 118)(35 117)(36 116)(37 115)(38 114)(39 113)(40 112)(41 146)(42 145)(43 144)(44 143)(45 142)(46 141)(47 140)(48 139)(49 138)(50 137)(51 136)(52 135)(53 134)(54 133)(55 132)(56 131)(57 130)(58 129)(59 128)(60 127)(61 126)(62 125)(63 124)(64 123)(65 122)(66 121)(67 160)(68 159)(69 158)(70 157)(71 156)(72 155)(73 154)(74 153)(75 152)(76 151)(77 150)(78 149)(79 148)(80 147)
G:=sub<Sym(160)| (1,58,97,155)(2,59,98,156)(3,60,99,157)(4,61,100,158)(5,62,101,159)(6,63,102,160)(7,64,103,121)(8,65,104,122)(9,66,105,123)(10,67,106,124)(11,68,107,125)(12,69,108,126)(13,70,109,127)(14,71,110,128)(15,72,111,129)(16,73,112,130)(17,74,113,131)(18,75,114,132)(19,76,115,133)(20,77,116,134)(21,78,117,135)(22,79,118,136)(23,80,119,137)(24,41,120,138)(25,42,81,139)(26,43,82,140)(27,44,83,141)(28,45,84,142)(29,46,85,143)(30,47,86,144)(31,48,87,145)(32,49,88,146)(33,50,89,147)(34,51,90,148)(35,52,91,149)(36,53,92,150)(37,54,93,151)(38,55,94,152)(39,56,95,153)(40,57,96,154), (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), (1,111)(2,110)(3,109)(4,108)(5,107)(6,106)(7,105)(8,104)(9,103)(10,102)(11,101)(12,100)(13,99)(14,98)(15,97)(16,96)(17,95)(18,94)(19,93)(20,92)(21,91)(22,90)(23,89)(24,88)(25,87)(26,86)(27,85)(28,84)(29,83)(30,82)(31,81)(32,120)(33,119)(34,118)(35,117)(36,116)(37,115)(38,114)(39,113)(40,112)(41,146)(42,145)(43,144)(44,143)(45,142)(46,141)(47,140)(48,139)(49,138)(50,137)(51,136)(52,135)(53,134)(54,133)(55,132)(56,131)(57,130)(58,129)(59,128)(60,127)(61,126)(62,125)(63,124)(64,123)(65,122)(66,121)(67,160)(68,159)(69,158)(70,157)(71,156)(72,155)(73,154)(74,153)(75,152)(76,151)(77,150)(78,149)(79,148)(80,147)>;
G:=Group( (1,58,97,155)(2,59,98,156)(3,60,99,157)(4,61,100,158)(5,62,101,159)(6,63,102,160)(7,64,103,121)(8,65,104,122)(9,66,105,123)(10,67,106,124)(11,68,107,125)(12,69,108,126)(13,70,109,127)(14,71,110,128)(15,72,111,129)(16,73,112,130)(17,74,113,131)(18,75,114,132)(19,76,115,133)(20,77,116,134)(21,78,117,135)(22,79,118,136)(23,80,119,137)(24,41,120,138)(25,42,81,139)(26,43,82,140)(27,44,83,141)(28,45,84,142)(29,46,85,143)(30,47,86,144)(31,48,87,145)(32,49,88,146)(33,50,89,147)(34,51,90,148)(35,52,91,149)(36,53,92,150)(37,54,93,151)(38,55,94,152)(39,56,95,153)(40,57,96,154), (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), (1,111)(2,110)(3,109)(4,108)(5,107)(6,106)(7,105)(8,104)(9,103)(10,102)(11,101)(12,100)(13,99)(14,98)(15,97)(16,96)(17,95)(18,94)(19,93)(20,92)(21,91)(22,90)(23,89)(24,88)(25,87)(26,86)(27,85)(28,84)(29,83)(30,82)(31,81)(32,120)(33,119)(34,118)(35,117)(36,116)(37,115)(38,114)(39,113)(40,112)(41,146)(42,145)(43,144)(44,143)(45,142)(46,141)(47,140)(48,139)(49,138)(50,137)(51,136)(52,135)(53,134)(54,133)(55,132)(56,131)(57,130)(58,129)(59,128)(60,127)(61,126)(62,125)(63,124)(64,123)(65,122)(66,121)(67,160)(68,159)(69,158)(70,157)(71,156)(72,155)(73,154)(74,153)(75,152)(76,151)(77,150)(78,149)(79,148)(80,147) );
G=PermutationGroup([(1,58,97,155),(2,59,98,156),(3,60,99,157),(4,61,100,158),(5,62,101,159),(6,63,102,160),(7,64,103,121),(8,65,104,122),(9,66,105,123),(10,67,106,124),(11,68,107,125),(12,69,108,126),(13,70,109,127),(14,71,110,128),(15,72,111,129),(16,73,112,130),(17,74,113,131),(18,75,114,132),(19,76,115,133),(20,77,116,134),(21,78,117,135),(22,79,118,136),(23,80,119,137),(24,41,120,138),(25,42,81,139),(26,43,82,140),(27,44,83,141),(28,45,84,142),(29,46,85,143),(30,47,86,144),(31,48,87,145),(32,49,88,146),(33,50,89,147),(34,51,90,148),(35,52,91,149),(36,53,92,150),(37,54,93,151),(38,55,94,152),(39,56,95,153),(40,57,96,154)], [(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)], [(1,111),(2,110),(3,109),(4,108),(5,107),(6,106),(7,105),(8,104),(9,103),(10,102),(11,101),(12,100),(13,99),(14,98),(15,97),(16,96),(17,95),(18,94),(19,93),(20,92),(21,91),(22,90),(23,89),(24,88),(25,87),(26,86),(27,85),(28,84),(29,83),(30,82),(31,81),(32,120),(33,119),(34,118),(35,117),(36,116),(37,115),(38,114),(39,113),(40,112),(41,146),(42,145),(43,144),(44,143),(45,142),(46,141),(47,140),(48,139),(49,138),(50,137),(51,136),(52,135),(53,134),(54,133),(55,132),(56,131),(57,130),(58,129),(59,128),(60,127),(61,126),(62,125),(63,124),(64,123),(65,122),(66,121),(67,160),(68,159),(69,158),(70,157),(71,156),(72,155),(73,154),(74,153),(75,152),(76,151),(77,150),(78,149),(79,148),(80,147)])
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