metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.19D4, C42.35D10, C4⋊C8⋊5D5, (C4×D20)⋊18C2, (C2×D40).4C2, (C2×C4).38D20, C4.131(D4×D5), (C2×C20).244D4, C20.340(C2×D4), (C2×C8).130D10, C5⋊3(D4.2D4), D20⋊5C4⋊12C2, C10.12(C4○D8), C20.44D4⋊8C2, C4.D20⋊12C2, (C4×C20).70C22, (C2×C40).23C22, C20.329(C4○D4), C2.18(C8⋊D10), C10.39(C4⋊D4), C2.12(C4⋊D20), C10.15(C8⋊C22), (C2×C20).754C23, C4.45(Q8⋊2D5), (C2×D20).17C22, C22.117(C2×D20), C2.14(D40⋊7C2), C4⋊Dic5.274C22, (C2×Dic10).17C22, (C5×C4⋊C8)⋊7C2, (C2×C40⋊C2)⋊19C2, (C2×C10).137(C2×D4), (C2×C4).699(C22×D5), SmallGroup(320,471)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.19D4
G = < a,b,c,d | a20=b2=c4=1, d2=a15, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a15b, dcd-1=a10c-1 >
Subgroups: 662 in 124 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C40, Dic10, C4×D5, D20, D20, C2×Dic5, C2×C20, C22×D5, D4.2D4, C40⋊C2, D40, C4⋊Dic5, D10⋊C4, C4×C20, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C20.44D4, D20⋊5C4, C5×C4⋊C8, C4×D20, C4.D20, C2×C40⋊C2, C2×D40, D20.19D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C4○D8, C8⋊C22, D20, C22×D5, D4.2D4, C2×D20, D4×D5, Q8⋊2D5, C4⋊D20, D40⋊7C2, C8⋊D10, D20.19D4
►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 38)(2 37)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 24)(16 23)(17 22)(18 21)(19 40)(20 39)(41 119)(42 118)(43 117)(44 116)(45 115)(46 114)(47 113)(48 112)(49 111)(50 110)(51 109)(52 108)(53 107)(54 106)(55 105)(56 104)(57 103)(58 102)(59 101)(60 120)(61 121)(62 140)(63 139)(64 138)(65 137)(66 136)(67 135)(68 134)(69 133)(70 132)(71 131)(72 130)(73 129)(74 128)(75 127)(76 126)(77 125)(78 124)(79 123)(80 122)(81 153)(82 152)(83 151)(84 150)(85 149)(86 148)(87 147)(88 146)(89 145)(90 144)(91 143)(92 142)(93 141)(94 160)(95 159)(96 158)(97 157)(98 156)(99 155)(100 154)
(1 64 48 86)(2 65 49 87)(3 66 50 88)(4 67 51 89)(5 68 52 90)(6 69 53 91)(7 70 54 92)(8 71 55 93)(9 72 56 94)(10 73 57 95)(11 74 58 96)(12 75 59 97)(13 76 60 98)(14 77 41 99)(15 78 42 100)(16 79 43 81)(17 80 44 82)(18 61 45 83)(19 62 46 84)(20 63 47 85)(21 121 115 151)(22 122 116 152)(23 123 117 153)(24 124 118 154)(25 125 119 155)(26 126 120 156)(27 127 101 157)(28 128 102 158)(29 129 103 159)(30 130 104 160)(31 131 105 141)(32 132 106 142)(33 133 107 143)(34 134 108 144)(35 135 109 145)(36 136 110 146)(37 137 111 147)(38 138 112 148)(39 139 113 149)(40 140 114 150)
(1 108 16 103 11 118 6 113)(2 109 17 104 12 119 7 114)(3 110 18 105 13 120 8 115)(4 111 19 106 14 101 9 116)(5 112 20 107 15 102 10 117)(21 50 36 45 31 60 26 55)(22 51 37 46 32 41 27 56)(23 52 38 47 33 42 28 57)(24 53 39 48 34 43 29 58)(25 54 40 49 35 44 30 59)(61 121 76 136 71 131 66 126)(62 122 77 137 72 132 67 127)(63 123 78 138 73 133 68 128)(64 124 79 139 74 134 69 129)(65 125 80 140 75 135 70 130)(81 149 96 144 91 159 86 154)(82 150 97 145 92 160 87 155)(83 151 98 146 93 141 88 156)(84 152 99 147 94 142 89 157)(85 153 100 148 95 143 90 158)
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), (1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,40)(20,39)(41,119)(42,118)(43,117)(44,116)(45,115)(46,114)(47,113)(48,112)(49,111)(50,110)(51,109)(52,108)(53,107)(54,106)(55,105)(56,104)(57,103)(58,102)(59,101)(60,120)(61,121)(62,140)(63,139)(64,138)(65,137)(66,136)(67,135)(68,134)(69,133)(70,132)(71,131)(72,130)(73,129)(74,128)(75,127)(76,126)(77,125)(78,124)(79,123)(80,122)(81,153)(82,152)(83,151)(84,150)(85,149)(86,148)(87,147)(88,146)(89,145)(90,144)(91,143)(92,142)(93,141)(94,160)(95,159)(96,158)(97,157)(98,156)(99,155)(100,154), (1,64,48,86)(2,65,49,87)(3,66,50,88)(4,67,51,89)(5,68,52,90)(6,69,53,91)(7,70,54,92)(8,71,55,93)(9,72,56,94)(10,73,57,95)(11,74,58,96)(12,75,59,97)(13,76,60,98)(14,77,41,99)(15,78,42,100)(16,79,43,81)(17,80,44,82)(18,61,45,83)(19,62,46,84)(20,63,47,85)(21,121,115,151)(22,122,116,152)(23,123,117,153)(24,124,118,154)(25,125,119,155)(26,126,120,156)(27,127,101,157)(28,128,102,158)(29,129,103,159)(30,130,104,160)(31,131,105,141)(32,132,106,142)(33,133,107,143)(34,134,108,144)(35,135,109,145)(36,136,110,146)(37,137,111,147)(38,138,112,148)(39,139,113,149)(40,140,114,150), (1,108,16,103,11,118,6,113)(2,109,17,104,12,119,7,114)(3,110,18,105,13,120,8,115)(4,111,19,106,14,101,9,116)(5,112,20,107,15,102,10,117)(21,50,36,45,31,60,26,55)(22,51,37,46,32,41,27,56)(23,52,38,47,33,42,28,57)(24,53,39,48,34,43,29,58)(25,54,40,49,35,44,30,59)(61,121,76,136,71,131,66,126)(62,122,77,137,72,132,67,127)(63,123,78,138,73,133,68,128)(64,124,79,139,74,134,69,129)(65,125,80,140,75,135,70,130)(81,149,96,144,91,159,86,154)(82,150,97,145,92,160,87,155)(83,151,98,146,93,141,88,156)(84,152,99,147,94,142,89,157)(85,153,100,148,95,143,90,158)>;
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), (1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,40)(20,39)(41,119)(42,118)(43,117)(44,116)(45,115)(46,114)(47,113)(48,112)(49,111)(50,110)(51,109)(52,108)(53,107)(54,106)(55,105)(56,104)(57,103)(58,102)(59,101)(60,120)(61,121)(62,140)(63,139)(64,138)(65,137)(66,136)(67,135)(68,134)(69,133)(70,132)(71,131)(72,130)(73,129)(74,128)(75,127)(76,126)(77,125)(78,124)(79,123)(80,122)(81,153)(82,152)(83,151)(84,150)(85,149)(86,148)(87,147)(88,146)(89,145)(90,144)(91,143)(92,142)(93,141)(94,160)(95,159)(96,158)(97,157)(98,156)(99,155)(100,154), (1,64,48,86)(2,65,49,87)(3,66,50,88)(4,67,51,89)(5,68,52,90)(6,69,53,91)(7,70,54,92)(8,71,55,93)(9,72,56,94)(10,73,57,95)(11,74,58,96)(12,75,59,97)(13,76,60,98)(14,77,41,99)(15,78,42,100)(16,79,43,81)(17,80,44,82)(18,61,45,83)(19,62,46,84)(20,63,47,85)(21,121,115,151)(22,122,116,152)(23,123,117,153)(24,124,118,154)(25,125,119,155)(26,126,120,156)(27,127,101,157)(28,128,102,158)(29,129,103,159)(30,130,104,160)(31,131,105,141)(32,132,106,142)(33,133,107,143)(34,134,108,144)(35,135,109,145)(36,136,110,146)(37,137,111,147)(38,138,112,148)(39,139,113,149)(40,140,114,150), (1,108,16,103,11,118,6,113)(2,109,17,104,12,119,7,114)(3,110,18,105,13,120,8,115)(4,111,19,106,14,101,9,116)(5,112,20,107,15,102,10,117)(21,50,36,45,31,60,26,55)(22,51,37,46,32,41,27,56)(23,52,38,47,33,42,28,57)(24,53,39,48,34,43,29,58)(25,54,40,49,35,44,30,59)(61,121,76,136,71,131,66,126)(62,122,77,137,72,132,67,127)(63,123,78,138,73,133,68,128)(64,124,79,139,74,134,69,129)(65,125,80,140,75,135,70,130)(81,149,96,144,91,159,86,154)(82,150,97,145,92,160,87,155)(83,151,98,146,93,141,88,156)(84,152,99,147,94,142,89,157)(85,153,100,148,95,143,90,158) );
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)], [(1,38),(2,37),(3,36),(4,35),(5,34),(6,33),(7,32),(8,31),(9,30),(10,29),(11,28),(12,27),(13,26),(14,25),(15,24),(16,23),(17,22),(18,21),(19,40),(20,39),(41,119),(42,118),(43,117),(44,116),(45,115),(46,114),(47,113),(48,112),(49,111),(50,110),(51,109),(52,108),(53,107),(54,106),(55,105),(56,104),(57,103),(58,102),(59,101),(60,120),(61,121),(62,140),(63,139),(64,138),(65,137),(66,136),(67,135),(68,134),(69,133),(70,132),(71,131),(72,130),(73,129),(74,128),(75,127),(76,126),(77,125),(78,124),(79,123),(80,122),(81,153),(82,152),(83,151),(84,150),(85,149),(86,148),(87,147),(88,146),(89,145),(90,144),(91,143),(92,142),(93,141),(94,160),(95,159),(96,158),(97,157),(98,156),(99,155),(100,154)], [(1,64,48,86),(2,65,49,87),(3,66,50,88),(4,67,51,89),(5,68,52,90),(6,69,53,91),(7,70,54,92),(8,71,55,93),(9,72,56,94),(10,73,57,95),(11,74,58,96),(12,75,59,97),(13,76,60,98),(14,77,41,99),(15,78,42,100),(16,79,43,81),(17,80,44,82),(18,61,45,83),(19,62,46,84),(20,63,47,85),(21,121,115,151),(22,122,116,152),(23,123,117,153),(24,124,118,154),(25,125,119,155),(26,126,120,156),(27,127,101,157),(28,128,102,158),(29,129,103,159),(30,130,104,160),(31,131,105,141),(32,132,106,142),(33,133,107,143),(34,134,108,144),(35,135,109,145),(36,136,110,146),(37,137,111,147),(38,138,112,148),(39,139,113,149),(40,140,114,150)], [(1,108,16,103,11,118,6,113),(2,109,17,104,12,119,7,114),(3,110,18,105,13,120,8,115),(4,111,19,106,14,101,9,116),(5,112,20,107,15,102,10,117),(21,50,36,45,31,60,26,55),(22,51,37,46,32,41,27,56),(23,52,38,47,33,42,28,57),(24,53,39,48,34,43,29,58),(25,54,40,49,35,44,30,59),(61,121,76,136,71,131,66,126),(62,122,77,137,72,132,67,127),(63,123,78,138,73,133,68,128),(64,124,79,139,74,134,69,129),(65,125,80,140,75,135,70,130),(81,149,96,144,91,159,86,154),(82,150,97,145,92,160,87,155),(83,151,98,146,93,141,88,156),(84,152,99,147,94,142,89,157),(85,153,100,148,95,143,90,158)]])
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 | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 40 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | C4○D8 | D20 | D40⋊7C2 | C8⋊C22 | D4×D5 | Q8⋊2D5 | C8⋊D10 |
| kernel | D20.19D4 | C20.44D4 | D20⋊5C4 | C5×C4⋊C8 | C4×D20 | C4.D20 | C2×C40⋊C2 | C2×D40 | D20 | C2×C20 | C4⋊C8 | C20 | C42 | C2×C8 | C10 | C2×C4 | C2 | C10 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 16 | 1 | 2 | 2 | 4 |
Matrix representation of D20.19D4 ►in GL4(𝔽41) generated by
| 30 | 39 | 0 | 0 |
| 16 | 14 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 38 | 8 | 0 | 0 |
| 40 | 3 | 0 | 0 |
| 0 | 0 | 32 | 23 |
| 0 | 0 | 9 | 9 |
| 32 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 40 | 39 |
| 0 | 0 | 1 | 1 |
| 20 | 33 | 0 | 0 |
| 23 | 38 | 0 | 0 |
| 0 | 0 | 32 | 23 |
| 0 | 0 | 0 | 9 |
G:=sub<GL(4,GF(41))| [30,16,0,0,39,14,0,0,0,0,40,0,0,0,0,40],[38,40,0,0,8,3,0,0,0,0,32,9,0,0,23,9],[32,0,0,0,0,32,0,0,0,0,40,1,0,0,39,1],[20,23,0,0,33,38,0,0,0,0,32,0,0,0,23,9] >;
D20.19D4 in GAP, Magma, Sage, TeX
D_{20}._{19}D_4 % in TeX
G:=Group("D20.19D4"); // GroupNames label
G:=SmallGroup(320,471);
// by ID
G=gap.SmallGroup(320,471);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,344,254,219,58,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=a^15,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=a^10*c^-1>;
// generators/relations