metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.7D20, Dic10.9D4, (C5×D4).2D4, C20.3(C2×D4), C4.87(D4×D5), C4.4(C2×D20), D4⋊C4⋊6D5, C4⋊C4.14D10, (C2×C8).10D10, D10⋊1C8⋊6C2, D10⋊2Q8⋊4C2, (C2×Dic20)⋊5C2, C10.Q16⋊7C2, C5⋊2(D4.7D4), C10.22C22≀C2, (C2×D4).137D10, C10.24(C4○D8), (C2×C40).10C22, (C22×D5).23D4, C22.181(D4×D5), C2.10(D8⋊3D5), (C2×C20).223C23, (C2×Dic5).199D4, (D4×C10).44C22, C2.25(C22⋊D20), C2.13(SD16⋊D5), C10.31(C8.C22), (C2×Dic10).64C22, (C2×D4.D5)⋊6C2, (C5×D4⋊C4)⋊6C2, (C2×C4×D5).19C22, (C2×D4⋊2D5).5C2, (C2×C10).236(C2×D4), (C5×C4⋊C4).24C22, (C2×C5⋊2C8).21C22, (C2×C4).330(C22×D5), SmallGroup(320,410)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D20
G = < a,b,c,d | a4=b2=c20=1, d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a-1b, dbd-1=ab, dcd-1=a2c-1 >
Subgroups: 638 in 152 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×C10, D4.7D4, Dic20, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, D4.D5, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, D4⋊2D5, C22×Dic5, C2×C5⋊D4, D4×C10, C10.Q16, D10⋊1C8, C5×D4⋊C4, D10⋊2Q8, C2×Dic20, C2×D4.D5, C2×D4⋊2D5, D4.D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C4○D8, C8.C22, D20, C22×D5, D4.7D4, C2×D20, D4×D5, C22⋊D20, D8⋊3D5, SD16⋊D5, D4.D20
►On 160 points
Generators in S160
(1 113 26 70)(2 71 27 114)(3 115 28 72)(4 73 29 116)(5 117 30 74)(6 75 31 118)(7 119 32 76)(8 77 33 120)(9 101 34 78)(10 79 35 102)(11 103 36 80)(12 61 37 104)(13 105 38 62)(14 63 39 106)(15 107 40 64)(16 65 21 108)(17 109 22 66)(18 67 23 110)(19 111 24 68)(20 69 25 112)(41 150 84 124)(42 125 85 151)(43 152 86 126)(44 127 87 153)(45 154 88 128)(46 129 89 155)(47 156 90 130)(48 131 91 157)(49 158 92 132)(50 133 93 159)(51 160 94 134)(52 135 95 141)(53 142 96 136)(54 137 97 143)(55 144 98 138)(56 139 99 145)(57 146 100 140)(58 121 81 147)(59 148 82 122)(60 123 83 149)
(1 49)(2 159)(3 51)(4 141)(5 53)(6 143)(7 55)(8 145)(9 57)(10 147)(11 59)(12 149)(13 41)(14 151)(15 43)(16 153)(17 45)(18 155)(19 47)(20 157)(21 127)(22 88)(23 129)(24 90)(25 131)(26 92)(27 133)(28 94)(29 135)(30 96)(31 137)(32 98)(33 139)(34 100)(35 121)(36 82)(37 123)(38 84)(39 125)(40 86)(42 106)(44 108)(46 110)(48 112)(50 114)(52 116)(54 118)(56 120)(58 102)(60 104)(61 83)(62 150)(63 85)(64 152)(65 87)(66 154)(67 89)(68 156)(69 91)(70 158)(71 93)(72 160)(73 95)(74 142)(75 97)(76 144)(77 99)(78 146)(79 81)(80 148)(101 140)(103 122)(105 124)(107 126)(109 128)(111 130)(113 132)(115 134)(117 136)(119 138)
(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 25 26 20)(2 19 27 24)(3 23 28 18)(4 17 29 22)(5 21 30 16)(6 15 31 40)(7 39 32 14)(8 13 33 38)(9 37 34 12)(10 11 35 36)(41 56 84 99)(42 98 85 55)(43 54 86 97)(44 96 87 53)(45 52 88 95)(46 94 89 51)(47 50 90 93)(48 92 91 49)(57 60 100 83)(58 82 81 59)(61 78 104 101)(62 120 105 77)(63 76 106 119)(64 118 107 75)(65 74 108 117)(66 116 109 73)(67 72 110 115)(68 114 111 71)(69 70 112 113)(79 80 102 103)(121 148 147 122)(123 146 149 140)(124 139 150 145)(125 144 151 138)(126 137 152 143)(127 142 153 136)(128 135 154 141)(129 160 155 134)(130 133 156 159)(131 158 157 132)
G:=sub<Sym(160)| (1,113,26,70)(2,71,27,114)(3,115,28,72)(4,73,29,116)(5,117,30,74)(6,75,31,118)(7,119,32,76)(8,77,33,120)(9,101,34,78)(10,79,35,102)(11,103,36,80)(12,61,37,104)(13,105,38,62)(14,63,39,106)(15,107,40,64)(16,65,21,108)(17,109,22,66)(18,67,23,110)(19,111,24,68)(20,69,25,112)(41,150,84,124)(42,125,85,151)(43,152,86,126)(44,127,87,153)(45,154,88,128)(46,129,89,155)(47,156,90,130)(48,131,91,157)(49,158,92,132)(50,133,93,159)(51,160,94,134)(52,135,95,141)(53,142,96,136)(54,137,97,143)(55,144,98,138)(56,139,99,145)(57,146,100,140)(58,121,81,147)(59,148,82,122)(60,123,83,149), (1,49)(2,159)(3,51)(4,141)(5,53)(6,143)(7,55)(8,145)(9,57)(10,147)(11,59)(12,149)(13,41)(14,151)(15,43)(16,153)(17,45)(18,155)(19,47)(20,157)(21,127)(22,88)(23,129)(24,90)(25,131)(26,92)(27,133)(28,94)(29,135)(30,96)(31,137)(32,98)(33,139)(34,100)(35,121)(36,82)(37,123)(38,84)(39,125)(40,86)(42,106)(44,108)(46,110)(48,112)(50,114)(52,116)(54,118)(56,120)(58,102)(60,104)(61,83)(62,150)(63,85)(64,152)(65,87)(66,154)(67,89)(68,156)(69,91)(70,158)(71,93)(72,160)(73,95)(74,142)(75,97)(76,144)(77,99)(78,146)(79,81)(80,148)(101,140)(103,122)(105,124)(107,126)(109,128)(111,130)(113,132)(115,134)(117,136)(119,138), (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), (1,25,26,20)(2,19,27,24)(3,23,28,18)(4,17,29,22)(5,21,30,16)(6,15,31,40)(7,39,32,14)(8,13,33,38)(9,37,34,12)(10,11,35,36)(41,56,84,99)(42,98,85,55)(43,54,86,97)(44,96,87,53)(45,52,88,95)(46,94,89,51)(47,50,90,93)(48,92,91,49)(57,60,100,83)(58,82,81,59)(61,78,104,101)(62,120,105,77)(63,76,106,119)(64,118,107,75)(65,74,108,117)(66,116,109,73)(67,72,110,115)(68,114,111,71)(69,70,112,113)(79,80,102,103)(121,148,147,122)(123,146,149,140)(124,139,150,145)(125,144,151,138)(126,137,152,143)(127,142,153,136)(128,135,154,141)(129,160,155,134)(130,133,156,159)(131,158,157,132)>;
G:=Group( (1,113,26,70)(2,71,27,114)(3,115,28,72)(4,73,29,116)(5,117,30,74)(6,75,31,118)(7,119,32,76)(8,77,33,120)(9,101,34,78)(10,79,35,102)(11,103,36,80)(12,61,37,104)(13,105,38,62)(14,63,39,106)(15,107,40,64)(16,65,21,108)(17,109,22,66)(18,67,23,110)(19,111,24,68)(20,69,25,112)(41,150,84,124)(42,125,85,151)(43,152,86,126)(44,127,87,153)(45,154,88,128)(46,129,89,155)(47,156,90,130)(48,131,91,157)(49,158,92,132)(50,133,93,159)(51,160,94,134)(52,135,95,141)(53,142,96,136)(54,137,97,143)(55,144,98,138)(56,139,99,145)(57,146,100,140)(58,121,81,147)(59,148,82,122)(60,123,83,149), (1,49)(2,159)(3,51)(4,141)(5,53)(6,143)(7,55)(8,145)(9,57)(10,147)(11,59)(12,149)(13,41)(14,151)(15,43)(16,153)(17,45)(18,155)(19,47)(20,157)(21,127)(22,88)(23,129)(24,90)(25,131)(26,92)(27,133)(28,94)(29,135)(30,96)(31,137)(32,98)(33,139)(34,100)(35,121)(36,82)(37,123)(38,84)(39,125)(40,86)(42,106)(44,108)(46,110)(48,112)(50,114)(52,116)(54,118)(56,120)(58,102)(60,104)(61,83)(62,150)(63,85)(64,152)(65,87)(66,154)(67,89)(68,156)(69,91)(70,158)(71,93)(72,160)(73,95)(74,142)(75,97)(76,144)(77,99)(78,146)(79,81)(80,148)(101,140)(103,122)(105,124)(107,126)(109,128)(111,130)(113,132)(115,134)(117,136)(119,138), (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), (1,25,26,20)(2,19,27,24)(3,23,28,18)(4,17,29,22)(5,21,30,16)(6,15,31,40)(7,39,32,14)(8,13,33,38)(9,37,34,12)(10,11,35,36)(41,56,84,99)(42,98,85,55)(43,54,86,97)(44,96,87,53)(45,52,88,95)(46,94,89,51)(47,50,90,93)(48,92,91,49)(57,60,100,83)(58,82,81,59)(61,78,104,101)(62,120,105,77)(63,76,106,119)(64,118,107,75)(65,74,108,117)(66,116,109,73)(67,72,110,115)(68,114,111,71)(69,70,112,113)(79,80,102,103)(121,148,147,122)(123,146,149,140)(124,139,150,145)(125,144,151,138)(126,137,152,143)(127,142,153,136)(128,135,154,141)(129,160,155,134)(130,133,156,159)(131,158,157,132) );
G=PermutationGroup([[(1,113,26,70),(2,71,27,114),(3,115,28,72),(4,73,29,116),(5,117,30,74),(6,75,31,118),(7,119,32,76),(8,77,33,120),(9,101,34,78),(10,79,35,102),(11,103,36,80),(12,61,37,104),(13,105,38,62),(14,63,39,106),(15,107,40,64),(16,65,21,108),(17,109,22,66),(18,67,23,110),(19,111,24,68),(20,69,25,112),(41,150,84,124),(42,125,85,151),(43,152,86,126),(44,127,87,153),(45,154,88,128),(46,129,89,155),(47,156,90,130),(48,131,91,157),(49,158,92,132),(50,133,93,159),(51,160,94,134),(52,135,95,141),(53,142,96,136),(54,137,97,143),(55,144,98,138),(56,139,99,145),(57,146,100,140),(58,121,81,147),(59,148,82,122),(60,123,83,149)], [(1,49),(2,159),(3,51),(4,141),(5,53),(6,143),(7,55),(8,145),(9,57),(10,147),(11,59),(12,149),(13,41),(14,151),(15,43),(16,153),(17,45),(18,155),(19,47),(20,157),(21,127),(22,88),(23,129),(24,90),(25,131),(26,92),(27,133),(28,94),(29,135),(30,96),(31,137),(32,98),(33,139),(34,100),(35,121),(36,82),(37,123),(38,84),(39,125),(40,86),(42,106),(44,108),(46,110),(48,112),(50,114),(52,116),(54,118),(56,120),(58,102),(60,104),(61,83),(62,150),(63,85),(64,152),(65,87),(66,154),(67,89),(68,156),(69,91),(70,158),(71,93),(72,160),(73,95),(74,142),(75,97),(76,144),(77,99),(78,146),(79,81),(80,148),(101,140),(103,122),(105,124),(107,126),(109,128),(111,130),(113,132),(115,134),(117,136),(119,138)], [(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)], [(1,25,26,20),(2,19,27,24),(3,23,28,18),(4,17,29,22),(5,21,30,16),(6,15,31,40),(7,39,32,14),(8,13,33,38),(9,37,34,12),(10,11,35,36),(41,56,84,99),(42,98,85,55),(43,54,86,97),(44,96,87,53),(45,52,88,95),(46,94,89,51),(47,50,90,93),(48,92,91,49),(57,60,100,83),(58,82,81,59),(61,78,104,101),(62,120,105,77),(63,76,106,119),(64,118,107,75),(65,74,108,117),(66,116,109,73),(67,72,110,115),(68,114,111,71),(69,70,112,113),(79,80,102,103),(121,148,147,122),(123,146,149,140),(124,139,150,145),(125,144,151,138),(126,137,152,143),(127,142,153,136),(128,135,154,141),(129,160,155,134),(130,133,156,159),(131,158,157,132)]])
47 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 | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| 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 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 2 | 2 | 8 | 10 | 10 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | C4○D8 | D20 | C8.C22 | D4×D5 | D4×D5 | D8⋊3D5 | SD16⋊D5 |
| kernel | D4.D20 | C10.Q16 | D10⋊1C8 | C5×D4⋊C4 | D10⋊2Q8 | C2×Dic20 | C2×D4.D5 | C2×D4⋊2D5 | Dic10 | C2×Dic5 | C5×D4 | C22×D5 | D4⋊C4 | C4⋊C4 | C2×C8 | C2×D4 | C10 | D4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D4.D20 ►in GL4(𝔽41) generated by
| 32 | 0 | 0 | 0 |
| 40 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 19 | 27 | 0 | 0 |
| 14 | 22 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 20 | 9 | 0 | 0 |
| 24 | 21 | 0 | 0 |
| 0 | 0 | 27 | 30 |
| 0 | 0 | 11 | 32 |
| 20 | 9 | 0 | 0 |
| 1 | 21 | 0 | 0 |
| 0 | 0 | 11 | 14 |
| 0 | 0 | 9 | 30 |
G:=sub<GL(4,GF(41))| [32,40,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[19,14,0,0,27,22,0,0,0,0,40,0,0,0,0,40],[20,24,0,0,9,21,0,0,0,0,27,11,0,0,30,32],[20,1,0,0,9,21,0,0,0,0,11,9,0,0,14,30] >;
D4.D20 in GAP, Magma, Sage, TeX
D_4.D_{20} % in TeX
G:=Group("D4.D20"); // GroupNames label
G:=SmallGroup(320,410);
// by ID
G=gap.SmallGroup(320,410);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,254,219,226,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=1,d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^-1*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^-1>;
// generators/relations