metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.1D20, C42.50D10, (C4×D4)⋊4D5, (D4×C20)⋊4C2, C20⋊3C8⋊24C2, (C5×D4).18D4, C20.18(C2×D4), C4.14(C2×D20), (C2×C20).61D4, C4⋊C4.244D10, C5⋊4(D4.2D4), D20⋊6C4⋊31C2, (C2×D4).191D10, C4.10(C4○D20), C10.89(C4○D8), C20.51(C4○D4), C4.D20⋊13C2, C10.Q16⋊29C2, (C4×C20).87C22, C2.13(C20⋊7D4), C10.65(C4⋊D4), C10.87(C8⋊C22), (C2×C20).338C23, (C2×D20).98C22, C2.9(D4.D10), (D4×C10).233C22, C2.11(D4.8D10), (C2×Dic10).103C22, (C2×D4⋊D5).5C2, (C2×D4.D5)⋊7C2, (C2×C10).469(C2×D4), (C2×C4).218(C5⋊D4), (C5×C4⋊C4).275C22, (C2×C5⋊2C8).94C22, (C2×C4).438(C22×D5), C22.150(C2×C5⋊D4), SmallGroup(320,643)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.1D20
G = < a,b,c,d | a4=b2=c20=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c-1 >
Subgroups: 502 in 124 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C5⋊2C8, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, D4.2D4, C2×C5⋊2C8, D10⋊C4, D4⋊D5, D4.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×D20, C22×C20, D4×C10, C20⋊3C8, D20⋊6C4, C10.Q16, C4.D20, C2×D4⋊D5, C2×D4.D5, D4×C20, D4.1D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C4○D8, C8⋊C22, D20, C5⋊D4, C22×D5, D4.2D4, C2×D20, C4○D20, C2×C5⋊D4, C20⋊7D4, D4.D10, D4.8D10, D4.1D20
►On 160 points
Generators in S160
(1 139 141 104)(2 140 142 105)(3 121 143 106)(4 122 144 107)(5 123 145 108)(6 124 146 109)(7 125 147 110)(8 126 148 111)(9 127 149 112)(10 128 150 113)(11 129 151 114)(12 130 152 115)(13 131 153 116)(14 132 154 117)(15 133 155 118)(16 134 156 119)(17 135 157 120)(18 136 158 101)(19 137 159 102)(20 138 160 103)(21 81 64 45)(22 82 65 46)(23 83 66 47)(24 84 67 48)(25 85 68 49)(26 86 69 50)(27 87 70 51)(28 88 71 52)(29 89 72 53)(30 90 73 54)(31 91 74 55)(32 92 75 56)(33 93 76 57)(34 94 77 58)(35 95 78 59)(36 96 79 60)(37 97 80 41)(38 98 61 42)(39 99 62 43)(40 100 63 44)
(1 60)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 47)(9 48)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)(21 109)(22 110)(23 111)(24 112)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(33 101)(34 102)(35 103)(36 104)(37 105)(38 106)(39 107)(40 108)(61 121)(62 122)(63 123)(64 124)(65 125)(66 126)(67 127)(68 128)(69 129)(70 130)(71 131)(72 132)(73 133)(74 134)(75 135)(76 136)(77 137)(78 138)(79 139)(80 140)(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)(96 141)(97 142)(98 143)(99 144)(100 145)
(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)
(1 160 141 20)(2 19 142 159)(3 158 143 18)(4 17 144 157)(5 156 145 16)(6 15 146 155)(7 154 147 14)(8 13 148 153)(9 152 149 12)(10 11 150 151)(21 90 64 54)(22 53 65 89)(23 88 66 52)(24 51 67 87)(25 86 68 50)(26 49 69 85)(27 84 70 48)(28 47 71 83)(29 82 72 46)(30 45 73 81)(31 100 74 44)(32 43 75 99)(33 98 76 42)(34 41 77 97)(35 96 78 60)(36 59 79 95)(37 94 80 58)(38 57 61 93)(39 92 62 56)(40 55 63 91)(101 121 136 106)(102 105 137 140)(103 139 138 104)(107 135 122 120)(108 119 123 134)(109 133 124 118)(110 117 125 132)(111 131 126 116)(112 115 127 130)(113 129 128 114)
G:=sub<Sym(160)| (1,139,141,104)(2,140,142,105)(3,121,143,106)(4,122,144,107)(5,123,145,108)(6,124,146,109)(7,125,147,110)(8,126,148,111)(9,127,149,112)(10,128,150,113)(11,129,151,114)(12,130,152,115)(13,131,153,116)(14,132,154,117)(15,133,155,118)(16,134,156,119)(17,135,157,120)(18,136,158,101)(19,137,159,102)(20,138,160,103)(21,81,64,45)(22,82,65,46)(23,83,66,47)(24,84,67,48)(25,85,68,49)(26,86,69,50)(27,87,70,51)(28,88,71,52)(29,89,72,53)(30,90,73,54)(31,91,74,55)(32,92,75,56)(33,93,76,57)(34,94,77,58)(35,95,78,59)(36,96,79,60)(37,97,80,41)(38,98,61,42)(39,99,62,43)(40,100,63,44), (1,60)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,127)(68,128)(69,129)(70,130)(71,131)(72,132)(73,133)(74,134)(75,135)(76,136)(77,137)(78,138)(79,139)(80,140)(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)(96,141)(97,142)(98,143)(99,144)(100,145), (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), (1,160,141,20)(2,19,142,159)(3,158,143,18)(4,17,144,157)(5,156,145,16)(6,15,146,155)(7,154,147,14)(8,13,148,153)(9,152,149,12)(10,11,150,151)(21,90,64,54)(22,53,65,89)(23,88,66,52)(24,51,67,87)(25,86,68,50)(26,49,69,85)(27,84,70,48)(28,47,71,83)(29,82,72,46)(30,45,73,81)(31,100,74,44)(32,43,75,99)(33,98,76,42)(34,41,77,97)(35,96,78,60)(36,59,79,95)(37,94,80,58)(38,57,61,93)(39,92,62,56)(40,55,63,91)(101,121,136,106)(102,105,137,140)(103,139,138,104)(107,135,122,120)(108,119,123,134)(109,133,124,118)(110,117,125,132)(111,131,126,116)(112,115,127,130)(113,129,128,114)>;
G:=Group( (1,139,141,104)(2,140,142,105)(3,121,143,106)(4,122,144,107)(5,123,145,108)(6,124,146,109)(7,125,147,110)(8,126,148,111)(9,127,149,112)(10,128,150,113)(11,129,151,114)(12,130,152,115)(13,131,153,116)(14,132,154,117)(15,133,155,118)(16,134,156,119)(17,135,157,120)(18,136,158,101)(19,137,159,102)(20,138,160,103)(21,81,64,45)(22,82,65,46)(23,83,66,47)(24,84,67,48)(25,85,68,49)(26,86,69,50)(27,87,70,51)(28,88,71,52)(29,89,72,53)(30,90,73,54)(31,91,74,55)(32,92,75,56)(33,93,76,57)(34,94,77,58)(35,95,78,59)(36,96,79,60)(37,97,80,41)(38,98,61,42)(39,99,62,43)(40,100,63,44), (1,60)(2,41)(3,42)(4,43)(5,44)(6,45)(7,46)(8,47)(9,48)(10,49)(11,50)(12,51)(13,52)(14,53)(15,54)(16,55)(17,56)(18,57)(19,58)(20,59)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,101)(34,102)(35,103)(36,104)(37,105)(38,106)(39,107)(40,108)(61,121)(62,122)(63,123)(64,124)(65,125)(66,126)(67,127)(68,128)(69,129)(70,130)(71,131)(72,132)(73,133)(74,134)(75,135)(76,136)(77,137)(78,138)(79,139)(80,140)(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)(96,141)(97,142)(98,143)(99,144)(100,145), (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), (1,160,141,20)(2,19,142,159)(3,158,143,18)(4,17,144,157)(5,156,145,16)(6,15,146,155)(7,154,147,14)(8,13,148,153)(9,152,149,12)(10,11,150,151)(21,90,64,54)(22,53,65,89)(23,88,66,52)(24,51,67,87)(25,86,68,50)(26,49,69,85)(27,84,70,48)(28,47,71,83)(29,82,72,46)(30,45,73,81)(31,100,74,44)(32,43,75,99)(33,98,76,42)(34,41,77,97)(35,96,78,60)(36,59,79,95)(37,94,80,58)(38,57,61,93)(39,92,62,56)(40,55,63,91)(101,121,136,106)(102,105,137,140)(103,139,138,104)(107,135,122,120)(108,119,123,134)(109,133,124,118)(110,117,125,132)(111,131,126,116)(112,115,127,130)(113,129,128,114) );
G=PermutationGroup([[(1,139,141,104),(2,140,142,105),(3,121,143,106),(4,122,144,107),(5,123,145,108),(6,124,146,109),(7,125,147,110),(8,126,148,111),(9,127,149,112),(10,128,150,113),(11,129,151,114),(12,130,152,115),(13,131,153,116),(14,132,154,117),(15,133,155,118),(16,134,156,119),(17,135,157,120),(18,136,158,101),(19,137,159,102),(20,138,160,103),(21,81,64,45),(22,82,65,46),(23,83,66,47),(24,84,67,48),(25,85,68,49),(26,86,69,50),(27,87,70,51),(28,88,71,52),(29,89,72,53),(30,90,73,54),(31,91,74,55),(32,92,75,56),(33,93,76,57),(34,94,77,58),(35,95,78,59),(36,96,79,60),(37,97,80,41),(38,98,61,42),(39,99,62,43),(40,100,63,44)], [(1,60),(2,41),(3,42),(4,43),(5,44),(6,45),(7,46),(8,47),(9,48),(10,49),(11,50),(12,51),(13,52),(14,53),(15,54),(16,55),(17,56),(18,57),(19,58),(20,59),(21,109),(22,110),(23,111),(24,112),(25,113),(26,114),(27,115),(28,116),(29,117),(30,118),(31,119),(32,120),(33,101),(34,102),(35,103),(36,104),(37,105),(38,106),(39,107),(40,108),(61,121),(62,122),(63,123),(64,124),(65,125),(66,126),(67,127),(68,128),(69,129),(70,130),(71,131),(72,132),(73,133),(74,134),(75,135),(76,136),(77,137),(78,138),(79,139),(80,140),(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),(96,141),(97,142),(98,143),(99,144),(100,145)], [(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)], [(1,160,141,20),(2,19,142,159),(3,158,143,18),(4,17,144,157),(5,156,145,16),(6,15,146,155),(7,154,147,14),(8,13,148,153),(9,152,149,12),(10,11,150,151),(21,90,64,54),(22,53,65,89),(23,88,66,52),(24,51,67,87),(25,86,68,50),(26,49,69,85),(27,84,70,48),(28,47,71,83),(29,82,72,46),(30,45,73,81),(31,100,74,44),(32,43,75,99),(33,98,76,42),(34,41,77,97),(35,96,78,60),(36,59,79,95),(37,94,80,58),(38,57,61,93),(39,92,62,56),(40,55,63,91),(101,121,136,106),(102,105,137,140),(103,139,138,104),(107,135,122,120),(108,119,123,134),(109,133,124,118),(110,117,125,132),(111,131,126,116),(112,115,127,130),(113,129,128,114)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 40 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 40 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | D20 | C4○D20 | C8⋊C22 | D4.D10 | D4.8D10 |
| kernel | D4.1D20 | C20⋊3C8 | D20⋊6C4 | C10.Q16 | C4.D20 | C2×D4⋊D5 | C2×D4.D5 | D4×C20 | C2×C20 | C5×D4 | C4×D4 | C20 | C42 | C4⋊C4 | C2×D4 | C10 | C2×C4 | D4 | C4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 1 | 4 | 4 |
Matrix representation of D4.1D20 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 21 |
| 0 | 0 | 37 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 35 |
| 0 | 0 | 34 | 0 |
| 16 | 30 | 0 | 0 |
| 27 | 2 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 39 | 30 | 0 | 0 |
| 4 | 2 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 36 | 9 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,37,0,0,21,40],[40,0,0,0,0,40,0,0,0,0,0,34,0,0,35,0],[16,27,0,0,30,2,0,0,0,0,32,0,0,0,0,32],[39,4,0,0,30,2,0,0,0,0,32,36,0,0,0,9] >;
D4.1D20 in GAP, Magma, Sage, TeX
D_4._1D_{20} % in TeX
G:=Group("D4.1D20"); // GroupNames label
G:=SmallGroup(320,643);
// by ID
G=gap.SmallGroup(320,643);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^-1>;
// generators/relations