metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊3D4, Dic5⋊2D8, (C2×D40)⋊5C2, C5⋊1(C4⋊D8), C2.11(D5×D8), C4.88(D4×D5), D4⋊C4⋊8D5, C20⋊D4⋊2C2, C10.25(C2×D8), (C2×C8).12D10, D20⋊8C4⋊3C2, D20⋊6C4⋊8C2, C4⋊C4.140D10, (C2×D4).31D10, C4.3(C4○D20), C20.112(C2×D4), C20.8Q8⋊7C2, C20.11(C4○D4), (C2×C40).12C22, (C2×Dic5).32D4, C22.183(D4×D5), C10.19(C4⋊D4), C2.13(D40⋊C2), C10.58(C8⋊C22), (C2×C20).226C23, (D4×C10).47C22, (C2×D20).58C22, C2.22(D10⋊D4), (C4×Dic5).19C22, (C2×D4⋊D5)⋊6C2, (C5×D4⋊C4)⋊8C2, (C2×C10).239(C2×D4), (C5×C4⋊C4).27C22, (C2×C5⋊2C8).24C22, (C2×C4).333(C22×D5), SmallGroup(320,413)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊3D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=dad=a-1, cac-1=a9, cbc-1=a8b, dbd=a3b, dcd=c-1 >
Subgroups: 758 in 140 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, C22×C4, C2×D4, C2×D4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, D4⋊C4, C4⋊C8, C4×D4, C4⋊1D4, C2×D8, C5⋊2C8, C40, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C4⋊D8, D40, C2×C5⋊2C8, C4×Dic5, D10⋊C4, D4⋊D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C2×C5⋊D4, D4×C10, D20⋊6C4, C20.8Q8, C5×D4⋊C4, D20⋊8C4, C2×D40, C2×D4⋊D5, C20⋊D4, D20⋊3D4
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, C4○D4, D10, C4⋊D4, C2×D8, C8⋊C22, C22×D5, C4⋊D8, C4○D20, D4×D5, D10⋊D4, D5×D8, D40⋊C2, D20⋊3D4
►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 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 60)(14 59)(15 58)(16 57)(17 56)(18 55)(19 54)(20 53)(21 102)(22 101)(23 120)(24 119)(25 118)(26 117)(27 116)(28 115)(29 114)(30 113)(31 112)(32 111)(33 110)(34 109)(35 108)(36 107)(37 106)(38 105)(39 104)(40 103)(61 89)(62 88)(63 87)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(70 100)(71 99)(72 98)(73 97)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(121 142)(122 141)(123 160)(124 159)(125 158)(126 157)(127 156)(128 155)(129 154)(130 153)(131 152)(132 151)(133 150)(134 149)(135 148)(136 147)(137 146)(138 145)(139 144)(140 143)
(1 76 39 143)(2 65 40 152)(3 74 21 141)(4 63 22 150)(5 72 23 159)(6 61 24 148)(7 70 25 157)(8 79 26 146)(9 68 27 155)(10 77 28 144)(11 66 29 153)(12 75 30 142)(13 64 31 151)(14 73 32 160)(15 62 33 149)(16 71 34 158)(17 80 35 147)(18 69 36 156)(19 78 37 145)(20 67 38 154)(41 87 113 133)(42 96 114 122)(43 85 115 131)(44 94 116 140)(45 83 117 129)(46 92 118 138)(47 81 119 127)(48 90 120 136)(49 99 101 125)(50 88 102 134)(51 97 103 123)(52 86 104 132)(53 95 105 121)(54 84 106 130)(55 93 107 139)(56 82 108 128)(57 91 109 137)(58 100 110 126)(59 89 111 135)(60 98 112 124)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(38 40)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 158)(62 157)(63 156)(64 155)(65 154)(66 153)(67 152)(68 151)(69 150)(70 149)(71 148)(72 147)(73 146)(74 145)(75 144)(76 143)(77 142)(78 141)(79 160)(80 159)(81 130)(82 129)(83 128)(84 127)(85 126)(86 125)(87 124)(88 123)(89 122)(90 121)(91 140)(92 139)(93 138)(94 137)(95 136)(96 135)(97 134)(98 133)(99 132)(100 131)(101 104)(102 103)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)
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), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,60)(14,59)(15,58)(16,57)(17,56)(18,55)(19,54)(20,53)(21,102)(22,101)(23,120)(24,119)(25,118)(26,117)(27,116)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(121,142)(122,141)(123,160)(124,159)(125,158)(126,157)(127,156)(128,155)(129,154)(130,153)(131,152)(132,151)(133,150)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,143), (1,76,39,143)(2,65,40,152)(3,74,21,141)(4,63,22,150)(5,72,23,159)(6,61,24,148)(7,70,25,157)(8,79,26,146)(9,68,27,155)(10,77,28,144)(11,66,29,153)(12,75,30,142)(13,64,31,151)(14,73,32,160)(15,62,33,149)(16,71,34,158)(17,80,35,147)(18,69,36,156)(19,78,37,145)(20,67,38,154)(41,87,113,133)(42,96,114,122)(43,85,115,131)(44,94,116,140)(45,83,117,129)(46,92,118,138)(47,81,119,127)(48,90,120,136)(49,99,101,125)(50,88,102,134)(51,97,103,123)(52,86,104,132)(53,95,105,121)(54,84,106,130)(55,93,107,139)(56,82,108,128)(57,91,109,137)(58,100,110,126)(59,89,111,135)(60,98,112,124), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(38,40)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,158)(62,157)(63,156)(64,155)(65,154)(66,153)(67,152)(68,151)(69,150)(70,149)(71,148)(72,147)(73,146)(74,145)(75,144)(76,143)(77,142)(78,141)(79,160)(80,159)(81,130)(82,129)(83,128)(84,127)(85,126)(86,125)(87,124)(88,123)(89,122)(90,121)(91,140)(92,139)(93,138)(94,137)(95,136)(96,135)(97,134)(98,133)(99,132)(100,131)(101,104)(102,103)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113)>;
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), (1,52)(2,51)(3,50)(4,49)(5,48)(6,47)(7,46)(8,45)(9,44)(10,43)(11,42)(12,41)(13,60)(14,59)(15,58)(16,57)(17,56)(18,55)(19,54)(20,53)(21,102)(22,101)(23,120)(24,119)(25,118)(26,117)(27,116)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90)(121,142)(122,141)(123,160)(124,159)(125,158)(126,157)(127,156)(128,155)(129,154)(130,153)(131,152)(132,151)(133,150)(134,149)(135,148)(136,147)(137,146)(138,145)(139,144)(140,143), (1,76,39,143)(2,65,40,152)(3,74,21,141)(4,63,22,150)(5,72,23,159)(6,61,24,148)(7,70,25,157)(8,79,26,146)(9,68,27,155)(10,77,28,144)(11,66,29,153)(12,75,30,142)(13,64,31,151)(14,73,32,160)(15,62,33,149)(16,71,34,158)(17,80,35,147)(18,69,36,156)(19,78,37,145)(20,67,38,154)(41,87,113,133)(42,96,114,122)(43,85,115,131)(44,94,116,140)(45,83,117,129)(46,92,118,138)(47,81,119,127)(48,90,120,136)(49,99,101,125)(50,88,102,134)(51,97,103,123)(52,86,104,132)(53,95,105,121)(54,84,106,130)(55,93,107,139)(56,82,108,128)(57,91,109,137)(58,100,110,126)(59,89,111,135)(60,98,112,124), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(38,40)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,158)(62,157)(63,156)(64,155)(65,154)(66,153)(67,152)(68,151)(69,150)(70,149)(71,148)(72,147)(73,146)(74,145)(75,144)(76,143)(77,142)(78,141)(79,160)(80,159)(81,130)(82,129)(83,128)(84,127)(85,126)(86,125)(87,124)(88,123)(89,122)(90,121)(91,140)(92,139)(93,138)(94,137)(95,136)(96,135)(97,134)(98,133)(99,132)(100,131)(101,104)(102,103)(105,120)(106,119)(107,118)(108,117)(109,116)(110,115)(111,114)(112,113) );
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)], [(1,52),(2,51),(3,50),(4,49),(5,48),(6,47),(7,46),(8,45),(9,44),(10,43),(11,42),(12,41),(13,60),(14,59),(15,58),(16,57),(17,56),(18,55),(19,54),(20,53),(21,102),(22,101),(23,120),(24,119),(25,118),(26,117),(27,116),(28,115),(29,114),(30,113),(31,112),(32,111),(33,110),(34,109),(35,108),(36,107),(37,106),(38,105),(39,104),(40,103),(61,89),(62,88),(63,87),(64,86),(65,85),(66,84),(67,83),(68,82),(69,81),(70,100),(71,99),(72,98),(73,97),(74,96),(75,95),(76,94),(77,93),(78,92),(79,91),(80,90),(121,142),(122,141),(123,160),(124,159),(125,158),(126,157),(127,156),(128,155),(129,154),(130,153),(131,152),(132,151),(133,150),(134,149),(135,148),(136,147),(137,146),(138,145),(139,144),(140,143)], [(1,76,39,143),(2,65,40,152),(3,74,21,141),(4,63,22,150),(5,72,23,159),(6,61,24,148),(7,70,25,157),(8,79,26,146),(9,68,27,155),(10,77,28,144),(11,66,29,153),(12,75,30,142),(13,64,31,151),(14,73,32,160),(15,62,33,149),(16,71,34,158),(17,80,35,147),(18,69,36,156),(19,78,37,145),(20,67,38,154),(41,87,113,133),(42,96,114,122),(43,85,115,131),(44,94,116,140),(45,83,117,129),(46,92,118,138),(47,81,119,127),(48,90,120,136),(49,99,101,125),(50,88,102,134),(51,97,103,123),(52,86,104,132),(53,95,105,121),(54,84,106,130),(55,93,107,139),(56,82,108,128),(57,91,109,137),(58,100,110,126),(59,89,111,135),(60,98,112,124)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,30),(38,40),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(61,158),(62,157),(63,156),(64,155),(65,154),(66,153),(67,152),(68,151),(69,150),(70,149),(71,148),(72,147),(73,146),(74,145),(75,144),(76,143),(77,142),(78,141),(79,160),(80,159),(81,130),(82,129),(83,128),(84,127),(85,126),(86,125),(87,124),(88,123),(89,122),(90,121),(91,140),(92,139),(93,138),(94,137),(95,136),(96,135),(97,134),(98,133),(99,132),(100,131),(101,104),(102,103),(105,120),(106,119),(107,118),(108,117),(109,116),(110,115),(111,114),(112,113)]])
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 | 8 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 10 | 10 | 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 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | C4○D4 | D10 | D10 | D10 | C4○D20 | C8⋊C22 | D4×D5 | D4×D5 | D5×D8 | D40⋊C2 |
| kernel | D20⋊3D4 | D20⋊6C4 | C20.8Q8 | C5×D4⋊C4 | D20⋊8C4 | C2×D40 | C2×D4⋊D5 | C20⋊D4 | D20 | C2×Dic5 | D4⋊C4 | Dic5 | C20 | C4⋊C4 | C2×C8 | C2×D4 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D20⋊3D4 ►in GL4(𝔽41) generated by
| 0 | 40 | 0 | 0 |
| 1 | 35 | 0 | 0 |
| 0 | 0 | 1 | 9 |
| 0 | 0 | 18 | 40 |
| 28 | 39 | 0 | 0 |
| 2 | 13 | 0 | 0 |
| 0 | 0 | 24 | 26 |
| 0 | 0 | 11 | 17 |
| 23 | 6 | 0 | 0 |
| 21 | 18 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 18 | 40 |
G:=sub<GL(4,GF(41))| [0,1,0,0,40,35,0,0,0,0,1,18,0,0,9,40],[28,2,0,0,39,13,0,0,0,0,24,11,0,0,26,17],[23,21,0,0,6,18,0,0,0,0,40,0,0,0,0,40],[1,6,0,0,0,40,0,0,0,0,1,18,0,0,0,40] >;
D20⋊3D4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_3D_4 % in TeX
G:=Group("D20:3D4"); // GroupNames label
G:=SmallGroup(320,413);
// by ID
G=gap.SmallGroup(320,413);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,219,1684,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^8*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations