metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊3D20, Dic10⋊3D4, (C5×D4)⋊2D4, C20.2(C2×D4), C4.86(D4×D5), C4.3(C2×D20), C4⋊D20⋊3C2, C5⋊2(D4⋊D4), C4⋊C4.12D10, D4⋊C4⋊13D5, C10.Q16⋊6C2, (C2×C8).117D10, D10⋊1C8⋊12C2, C10.21C22≀C2, (C2×D4).136D10, C10.42(C4○D8), (C22×D5).21D4, C22.179(D4×D5), C2.17(D8⋊D5), C10.35(C8⋊C22), (C2×C40).128C22, (C2×C20).221C23, (C2×Dic5).198D4, (D4×C10).42C22, (C2×D20).55C22, C2.24(C22⋊D20), C2.12(SD16⋊3D5), (C2×Dic10).63C22, (C2×D4⋊D5)⋊4C2, (C2×D4⋊2D5)⋊1C2, (C2×C40⋊C2)⋊15C2, (C5×D4⋊C4)⋊13C2, (C2×C4×D5).17C22, (C2×C10).234(C2×D4), (C5×C4⋊C4).22C22, (C2×C5⋊2C8).19C22, (C2×C4).328(C22×D5), SmallGroup(320,408)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊3D20
G = < a,b,c,d | a4=b2=c20=d2=1, bab=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=c-1 >
Subgroups: 782 in 162 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, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C5⋊2C8, C40, Dic10, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C22×D5, C22×D5, C22×C10, D4⋊D4, C40⋊C2, C2×C5⋊2C8, D10⋊C4, D4⋊D5, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C2×D20, D4⋊2D5, C22×Dic5, C2×C5⋊D4, D4×C10, C10.Q16, D10⋊1C8, C5×D4⋊C4, C4⋊D20, C2×C40⋊C2, C2×D4⋊D5, C2×D4⋊2D5, D4⋊3D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C4○D8, C8⋊C22, D20, C22×D5, D4⋊D4, C2×D20, D4×D5, C22⋊D20, D8⋊D5, SD16⋊3D5, D4⋊3D20
►On 160 points
Generators in S160
(1 55 28 88)(2 89 29 56)(3 57 30 90)(4 91 31 58)(5 59 32 92)(6 93 33 60)(7 41 34 94)(8 95 35 42)(9 43 36 96)(10 97 37 44)(11 45 38 98)(12 99 39 46)(13 47 40 100)(14 81 21 48)(15 49 22 82)(16 83 23 50)(17 51 24 84)(18 85 25 52)(19 53 26 86)(20 87 27 54)(61 127 158 114)(62 115 159 128)(63 129 160 116)(64 117 141 130)(65 131 142 118)(66 119 143 132)(67 133 144 120)(68 101 145 134)(69 135 146 102)(70 103 147 136)(71 137 148 104)(72 105 149 138)(73 139 150 106)(74 107 151 140)(75 121 152 108)(76 109 153 122)(77 123 154 110)(78 111 155 124)(79 125 156 112)(80 113 157 126)
(1 148)(2 105)(3 150)(4 107)(5 152)(6 109)(7 154)(8 111)(9 156)(10 113)(11 158)(12 115)(13 160)(14 117)(15 142)(16 119)(17 144)(18 101)(19 146)(20 103)(21 130)(22 65)(23 132)(24 67)(25 134)(26 69)(27 136)(28 71)(29 138)(30 73)(31 140)(32 75)(33 122)(34 77)(35 124)(36 79)(37 126)(38 61)(39 128)(40 63)(41 123)(42 155)(43 125)(44 157)(45 127)(46 159)(47 129)(48 141)(49 131)(50 143)(51 133)(52 145)(53 135)(54 147)(55 137)(56 149)(57 139)(58 151)(59 121)(60 153)(62 99)(64 81)(66 83)(68 85)(70 87)(72 89)(74 91)(76 93)(78 95)(80 97)(82 118)(84 120)(86 102)(88 104)(90 106)(92 108)(94 110)(96 112)(98 114)(100 116)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 29)(22 28)(23 27)(24 26)(30 40)(31 39)(32 38)(33 37)(34 36)(41 96)(42 95)(43 94)(44 93)(45 92)(46 91)(47 90)(48 89)(49 88)(50 87)(51 86)(52 85)(53 84)(54 83)(55 82)(56 81)(57 100)(58 99)(59 98)(60 97)(61 108)(62 107)(63 106)(64 105)(65 104)(66 103)(67 102)(68 101)(69 120)(70 119)(71 118)(72 117)(73 116)(74 115)(75 114)(76 113)(77 112)(78 111)(79 110)(80 109)(121 158)(122 157)(123 156)(124 155)(125 154)(126 153)(127 152)(128 151)(129 150)(130 149)(131 148)(132 147)(133 146)(134 145)(135 144)(136 143)(137 142)(138 141)(139 160)(140 159)
G:=sub<Sym(160)| (1,55,28,88)(2,89,29,56)(3,57,30,90)(4,91,31,58)(5,59,32,92)(6,93,33,60)(7,41,34,94)(8,95,35,42)(9,43,36,96)(10,97,37,44)(11,45,38,98)(12,99,39,46)(13,47,40,100)(14,81,21,48)(15,49,22,82)(16,83,23,50)(17,51,24,84)(18,85,25,52)(19,53,26,86)(20,87,27,54)(61,127,158,114)(62,115,159,128)(63,129,160,116)(64,117,141,130)(65,131,142,118)(66,119,143,132)(67,133,144,120)(68,101,145,134)(69,135,146,102)(70,103,147,136)(71,137,148,104)(72,105,149,138)(73,139,150,106)(74,107,151,140)(75,121,152,108)(76,109,153,122)(77,123,154,110)(78,111,155,124)(79,125,156,112)(80,113,157,126), (1,148)(2,105)(3,150)(4,107)(5,152)(6,109)(7,154)(8,111)(9,156)(10,113)(11,158)(12,115)(13,160)(14,117)(15,142)(16,119)(17,144)(18,101)(19,146)(20,103)(21,130)(22,65)(23,132)(24,67)(25,134)(26,69)(27,136)(28,71)(29,138)(30,73)(31,140)(32,75)(33,122)(34,77)(35,124)(36,79)(37,126)(38,61)(39,128)(40,63)(41,123)(42,155)(43,125)(44,157)(45,127)(46,159)(47,129)(48,141)(49,131)(50,143)(51,133)(52,145)(53,135)(54,147)(55,137)(56,149)(57,139)(58,151)(59,121)(60,153)(62,99)(64,81)(66,83)(68,85)(70,87)(72,89)(74,91)(76,93)(78,95)(80,97)(82,118)(84,120)(86,102)(88,104)(90,106)(92,108)(94,110)(96,112)(98,114)(100,116), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,29)(22,28)(23,27)(24,26)(30,40)(31,39)(32,38)(33,37)(34,36)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,100)(58,99)(59,98)(60,97)(61,108)(62,107)(63,106)(64,105)(65,104)(66,103)(67,102)(68,101)(69,120)(70,119)(71,118)(72,117)(73,116)(74,115)(75,114)(76,113)(77,112)(78,111)(79,110)(80,109)(121,158)(122,157)(123,156)(124,155)(125,154)(126,153)(127,152)(128,151)(129,150)(130,149)(131,148)(132,147)(133,146)(134,145)(135,144)(136,143)(137,142)(138,141)(139,160)(140,159)>;
G:=Group( (1,55,28,88)(2,89,29,56)(3,57,30,90)(4,91,31,58)(5,59,32,92)(6,93,33,60)(7,41,34,94)(8,95,35,42)(9,43,36,96)(10,97,37,44)(11,45,38,98)(12,99,39,46)(13,47,40,100)(14,81,21,48)(15,49,22,82)(16,83,23,50)(17,51,24,84)(18,85,25,52)(19,53,26,86)(20,87,27,54)(61,127,158,114)(62,115,159,128)(63,129,160,116)(64,117,141,130)(65,131,142,118)(66,119,143,132)(67,133,144,120)(68,101,145,134)(69,135,146,102)(70,103,147,136)(71,137,148,104)(72,105,149,138)(73,139,150,106)(74,107,151,140)(75,121,152,108)(76,109,153,122)(77,123,154,110)(78,111,155,124)(79,125,156,112)(80,113,157,126), (1,148)(2,105)(3,150)(4,107)(5,152)(6,109)(7,154)(8,111)(9,156)(10,113)(11,158)(12,115)(13,160)(14,117)(15,142)(16,119)(17,144)(18,101)(19,146)(20,103)(21,130)(22,65)(23,132)(24,67)(25,134)(26,69)(27,136)(28,71)(29,138)(30,73)(31,140)(32,75)(33,122)(34,77)(35,124)(36,79)(37,126)(38,61)(39,128)(40,63)(41,123)(42,155)(43,125)(44,157)(45,127)(46,159)(47,129)(48,141)(49,131)(50,143)(51,133)(52,145)(53,135)(54,147)(55,137)(56,149)(57,139)(58,151)(59,121)(60,153)(62,99)(64,81)(66,83)(68,85)(70,87)(72,89)(74,91)(76,93)(78,95)(80,97)(82,118)(84,120)(86,102)(88,104)(90,106)(92,108)(94,110)(96,112)(98,114)(100,116), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,29)(22,28)(23,27)(24,26)(30,40)(31,39)(32,38)(33,37)(34,36)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,100)(58,99)(59,98)(60,97)(61,108)(62,107)(63,106)(64,105)(65,104)(66,103)(67,102)(68,101)(69,120)(70,119)(71,118)(72,117)(73,116)(74,115)(75,114)(76,113)(77,112)(78,111)(79,110)(80,109)(121,158)(122,157)(123,156)(124,155)(125,154)(126,153)(127,152)(128,151)(129,150)(130,149)(131,148)(132,147)(133,146)(134,145)(135,144)(136,143)(137,142)(138,141)(139,160)(140,159) );
G=PermutationGroup([[(1,55,28,88),(2,89,29,56),(3,57,30,90),(4,91,31,58),(5,59,32,92),(6,93,33,60),(7,41,34,94),(8,95,35,42),(9,43,36,96),(10,97,37,44),(11,45,38,98),(12,99,39,46),(13,47,40,100),(14,81,21,48),(15,49,22,82),(16,83,23,50),(17,51,24,84),(18,85,25,52),(19,53,26,86),(20,87,27,54),(61,127,158,114),(62,115,159,128),(63,129,160,116),(64,117,141,130),(65,131,142,118),(66,119,143,132),(67,133,144,120),(68,101,145,134),(69,135,146,102),(70,103,147,136),(71,137,148,104),(72,105,149,138),(73,139,150,106),(74,107,151,140),(75,121,152,108),(76,109,153,122),(77,123,154,110),(78,111,155,124),(79,125,156,112),(80,113,157,126)], [(1,148),(2,105),(3,150),(4,107),(5,152),(6,109),(7,154),(8,111),(9,156),(10,113),(11,158),(12,115),(13,160),(14,117),(15,142),(16,119),(17,144),(18,101),(19,146),(20,103),(21,130),(22,65),(23,132),(24,67),(25,134),(26,69),(27,136),(28,71),(29,138),(30,73),(31,140),(32,75),(33,122),(34,77),(35,124),(36,79),(37,126),(38,61),(39,128),(40,63),(41,123),(42,155),(43,125),(44,157),(45,127),(46,159),(47,129),(48,141),(49,131),(50,143),(51,133),(52,145),(53,135),(54,147),(55,137),(56,149),(57,139),(58,151),(59,121),(60,153),(62,99),(64,81),(66,83),(68,85),(70,87),(72,89),(74,91),(76,93),(78,95),(80,97),(82,118),(84,120),(86,102),(88,104),(90,106),(92,108),(94,110),(96,112),(98,114),(100,116)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,29),(22,28),(23,27),(24,26),(30,40),(31,39),(32,38),(33,37),(34,36),(41,96),(42,95),(43,94),(44,93),(45,92),(46,91),(47,90),(48,89),(49,88),(50,87),(51,86),(52,85),(53,84),(54,83),(55,82),(56,81),(57,100),(58,99),(59,98),(60,97),(61,108),(62,107),(63,106),(64,105),(65,104),(66,103),(67,102),(68,101),(69,120),(70,119),(71,118),(72,117),(73,116),(74,115),(75,114),(76,113),(77,112),(78,111),(79,110),(80,109),(121,158),(122,157),(123,156),(124,155),(125,154),(126,153),(127,152),(128,151),(129,150),(130,149),(131,148),(132,147),(133,146),(134,145),(135,144),(136,143),(137,142),(138,141),(139,160),(140,159)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 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 | 2 | 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 | 40 | 2 | 2 | 8 | 10 | 10 | 20 | 20 | 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⋊D5 | SD16⋊3D5 |
| kernel | D4⋊3D20 | C10.Q16 | D10⋊1C8 | C5×D4⋊C4 | C4⋊D20 | C2×C40⋊C2 | 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⋊3D20 ►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 |
| 2 | 30 | 0 | 0 |
| 27 | 16 | 0 | 0 |
| 0 | 0 | 9 | 25 |
| 0 | 0 | 0 | 32 |
| 40 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 4 | 1 |
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],[2,27,0,0,30,16,0,0,0,0,9,0,0,0,25,32],[40,5,0,0,0,1,0,0,0,0,40,4,0,0,0,1] >;
D4⋊3D20 in GAP, Magma, Sage, TeX
D_4\rtimes_3D_{20} % in TeX
G:=Group("D4:3D20"); // GroupNames label
G:=SmallGroup(320,408);
// by ID
G=gap.SmallGroup(320,408);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,254,219,58,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^20=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations