direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C40, C4⋊1(C2×C40), (C4×C40)⋊9C2, (C4×C8)⋊4C10, C4⋊C8⋊18C10, C20⋊12(C2×C8), C2.3(D4×C20), C4⋊C4.11C20, C22⋊1(C2×C40), (C22×C40)⋊9C2, (C22×C8)⋊5C10, C4.79(D4×C10), C22⋊C8⋊15C10, (D4×C20).29C2, (C2×D4).11C20, (D4×C10).36C4, (C4×D4).14C10, C10.141(C4×D4), C20.484(C2×D4), C22⋊C4.7C20, C2.4(C22×C40), C10.72(C8○D4), C42.68(C2×C10), C23.18(C2×C20), C10.57(C22×C8), C20.353(C4○D4), (C2×C40).361C22, (C2×C20).990C23, (C4×C20).353C22, C22.22(C22×C20), (C22×C20).499C22, (C5×C4⋊C8)⋊37C2, C2.2(C5×C8○D4), (C2×C10)⋊10(C2×C8), (C5×C4⋊C4).36C4, C4.51(C5×C4○D4), (C5×C22⋊C8)⋊32C2, (C2×C4).36(C2×C20), (C2×C20).385(C2×C4), (C2×C8).107(C2×C10), (C5×C22⋊C4).22C4, (C22×C4).95(C2×C10), (C2×C10).341(C22×C4), (C2×C4).158(C22×C10), (C22×C10).153(C2×C4), SmallGroup(320,935)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C40
G = < a,b,c | a40=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 178 in 134 conjugacy classes, 90 normal (38 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, C22×C4, C2×D4, C20, C20, C20, C2×C10, C2×C10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C40, C40, C2×C20, C2×C20, C2×C20, C5×D4, C22×C10, C8×D4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C2×C40, C2×C40, C22×C20, D4×C10, C4×C40, C5×C22⋊C8, C5×C4⋊C8, D4×C20, C22×C40, D4×C40
Quotients: C1, C2, C4, C22, C5, C8, C2×C4, D4, C23, C10, C2×C8, C22×C4, C2×D4, C4○D4, C20, C2×C10, C4×D4, C22×C8, C8○D4, C40, C2×C20, C5×D4, C22×C10, C8×D4, C2×C40, C22×C20, D4×C10, C5×C4○D4, D4×C20, C22×C40, C5×C8○D4, D4×C40
►On 160 points
Generators in S160
(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 59 135 117)(2 60 136 118)(3 61 137 119)(4 62 138 120)(5 63 139 81)(6 64 140 82)(7 65 141 83)(8 66 142 84)(9 67 143 85)(10 68 144 86)(11 69 145 87)(12 70 146 88)(13 71 147 89)(14 72 148 90)(15 73 149 91)(16 74 150 92)(17 75 151 93)(18 76 152 94)(19 77 153 95)(20 78 154 96)(21 79 155 97)(22 80 156 98)(23 41 157 99)(24 42 158 100)(25 43 159 101)(26 44 160 102)(27 45 121 103)(28 46 122 104)(29 47 123 105)(30 48 124 106)(31 49 125 107)(32 50 126 108)(33 51 127 109)(34 52 128 110)(35 53 129 111)(36 54 130 112)(37 55 131 113)(38 56 132 114)(39 57 133 115)(40 58 134 116)
(1 117)(2 118)(3 119)(4 120)(5 81)(6 82)(7 83)(8 84)(9 85)(10 86)(11 87)(12 88)(13 89)(14 90)(15 91)(16 92)(17 93)(18 94)(19 95)(20 96)(21 97)(22 98)(23 99)(24 100)(25 101)(26 102)(27 103)(28 104)(29 105)(30 106)(31 107)(32 108)(33 109)(34 110)(35 111)(36 112)(37 113)(38 114)(39 115)(40 116)(41 157)(42 158)(43 159)(44 160)(45 121)(46 122)(47 123)(48 124)(49 125)(50 126)(51 127)(52 128)(53 129)(54 130)(55 131)(56 132)(57 133)(58 134)(59 135)(60 136)(61 137)(62 138)(63 139)(64 140)(65 141)(66 142)(67 143)(68 144)(69 145)(70 146)(71 147)(72 148)(73 149)(74 150)(75 151)(76 152)(77 153)(78 154)(79 155)(80 156)
G:=sub<Sym(160)| (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), (1,59,135,117)(2,60,136,118)(3,61,137,119)(4,62,138,120)(5,63,139,81)(6,64,140,82)(7,65,141,83)(8,66,142,84)(9,67,143,85)(10,68,144,86)(11,69,145,87)(12,70,146,88)(13,71,147,89)(14,72,148,90)(15,73,149,91)(16,74,150,92)(17,75,151,93)(18,76,152,94)(19,77,153,95)(20,78,154,96)(21,79,155,97)(22,80,156,98)(23,41,157,99)(24,42,158,100)(25,43,159,101)(26,44,160,102)(27,45,121,103)(28,46,122,104)(29,47,123,105)(30,48,124,106)(31,49,125,107)(32,50,126,108)(33,51,127,109)(34,52,128,110)(35,53,129,111)(36,54,130,112)(37,55,131,113)(38,56,132,114)(39,57,133,115)(40,58,134,116), (1,117)(2,118)(3,119)(4,120)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,109)(34,110)(35,111)(36,112)(37,113)(38,114)(39,115)(40,116)(41,157)(42,158)(43,159)(44,160)(45,121)(46,122)(47,123)(48,124)(49,125)(50,126)(51,127)(52,128)(53,129)(54,130)(55,131)(56,132)(57,133)(58,134)(59,135)(60,136)(61,137)(62,138)(63,139)(64,140)(65,141)(66,142)(67,143)(68,144)(69,145)(70,146)(71,147)(72,148)(73,149)(74,150)(75,151)(76,152)(77,153)(78,154)(79,155)(80,156)>;
G:=Group( (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), (1,59,135,117)(2,60,136,118)(3,61,137,119)(4,62,138,120)(5,63,139,81)(6,64,140,82)(7,65,141,83)(8,66,142,84)(9,67,143,85)(10,68,144,86)(11,69,145,87)(12,70,146,88)(13,71,147,89)(14,72,148,90)(15,73,149,91)(16,74,150,92)(17,75,151,93)(18,76,152,94)(19,77,153,95)(20,78,154,96)(21,79,155,97)(22,80,156,98)(23,41,157,99)(24,42,158,100)(25,43,159,101)(26,44,160,102)(27,45,121,103)(28,46,122,104)(29,47,123,105)(30,48,124,106)(31,49,125,107)(32,50,126,108)(33,51,127,109)(34,52,128,110)(35,53,129,111)(36,54,130,112)(37,55,131,113)(38,56,132,114)(39,57,133,115)(40,58,134,116), (1,117)(2,118)(3,119)(4,120)(5,81)(6,82)(7,83)(8,84)(9,85)(10,86)(11,87)(12,88)(13,89)(14,90)(15,91)(16,92)(17,93)(18,94)(19,95)(20,96)(21,97)(22,98)(23,99)(24,100)(25,101)(26,102)(27,103)(28,104)(29,105)(30,106)(31,107)(32,108)(33,109)(34,110)(35,111)(36,112)(37,113)(38,114)(39,115)(40,116)(41,157)(42,158)(43,159)(44,160)(45,121)(46,122)(47,123)(48,124)(49,125)(50,126)(51,127)(52,128)(53,129)(54,130)(55,131)(56,132)(57,133)(58,134)(59,135)(60,136)(61,137)(62,138)(63,139)(64,140)(65,141)(66,142)(67,143)(68,144)(69,145)(70,146)(71,147)(72,148)(73,149)(74,150)(75,151)(76,152)(77,153)(78,154)(79,155)(80,156) );
G=PermutationGroup([[(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)], [(1,59,135,117),(2,60,136,118),(3,61,137,119),(4,62,138,120),(5,63,139,81),(6,64,140,82),(7,65,141,83),(8,66,142,84),(9,67,143,85),(10,68,144,86),(11,69,145,87),(12,70,146,88),(13,71,147,89),(14,72,148,90),(15,73,149,91),(16,74,150,92),(17,75,151,93),(18,76,152,94),(19,77,153,95),(20,78,154,96),(21,79,155,97),(22,80,156,98),(23,41,157,99),(24,42,158,100),(25,43,159,101),(26,44,160,102),(27,45,121,103),(28,46,122,104),(29,47,123,105),(30,48,124,106),(31,49,125,107),(32,50,126,108),(33,51,127,109),(34,52,128,110),(35,53,129,111),(36,54,130,112),(37,55,131,113),(38,56,132,114),(39,57,133,115),(40,58,134,116)], [(1,117),(2,118),(3,119),(4,120),(5,81),(6,82),(7,83),(8,84),(9,85),(10,86),(11,87),(12,88),(13,89),(14,90),(15,91),(16,92),(17,93),(18,94),(19,95),(20,96),(21,97),(22,98),(23,99),(24,100),(25,101),(26,102),(27,103),(28,104),(29,105),(30,106),(31,107),(32,108),(33,109),(34,110),(35,111),(36,112),(37,113),(38,114),(39,115),(40,116),(41,157),(42,158),(43,159),(44,160),(45,121),(46,122),(47,123),(48,124),(49,125),(50,126),(51,127),(52,128),(53,129),(54,130),(55,131),(56,132),(57,133),(58,134),(59,135),(60,136),(61,137),(62,138),(63,139),(64,140),(65,141),(66,142),(67,143),(68,144),(69,145),(70,146),(71,147),(72,148),(73,149),(74,150),(75,151),(76,152),(77,153),(78,154),(79,155),(80,156)]])
200 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 8I | ··· | 8T | 10A | ··· | 10L | 10M | ··· | 10AB | 20A | ··· | 20P | 20Q | ··· | 20AV | 40A | ··· | 40AF | 40AG | ··· | 40CB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C5 | C8 | C10 | C10 | C10 | C10 | C10 | C20 | C20 | C20 | C40 | D4 | C4○D4 | C8○D4 | C5×D4 | C5×C4○D4 | C5×C8○D4 |
| kernel | D4×C40 | C4×C40 | C5×C22⋊C8 | C5×C4⋊C8 | D4×C20 | C22×C40 | C5×C22⋊C4 | C5×C4⋊C4 | D4×C10 | C8×D4 | C5×D4 | C4×C8 | C22⋊C8 | C4⋊C8 | C4×D4 | C22×C8 | C22⋊C4 | C4⋊C4 | C2×D4 | D4 | C40 | C20 | C10 | C8 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 16 | 4 | 8 | 4 | 4 | 8 | 16 | 8 | 8 | 64 | 2 | 2 | 4 | 8 | 8 | 16 |
Matrix representation of D4×C40 ►in GL3(𝔽41) generated by
| 38 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 40 | 0 | 0 |
| 0 | 1 | 18 |
| 0 | 9 | 40 |
| 1 | 0 | 0 |
| 0 | 1 | 18 |
| 0 | 0 | 40 |
G:=sub<GL(3,GF(41))| [38,0,0,0,2,0,0,0,2],[40,0,0,0,1,9,0,18,40],[1,0,0,0,1,0,0,18,40] >;
D4×C40 in GAP, Magma, Sage, TeX
D_4\times C_{40} % in TeX
G:=Group("D4xC40"); // GroupNames label
G:=SmallGroup(320,935);
// by ID
G=gap.SmallGroup(320,935);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,436,124]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations