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 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×2], C2×C4 [×3], C2×C4 [×4], D4 [×5], Q8 [×2], C23 [×2], D5 [×3], C10 [×3], C42, C22⋊C4 [×3], C4⋊C4, C2×C8 [×2], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], D10 [×7], C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C40 [×2], Dic10 [×2], C4×D5 [×2], D20 [×2], D20 [×3], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], D4.2D4, C40⋊C2 [×2], D40 [×2], C4⋊Dic5, D10⋊C4 [×3], C4×C20, C2×C40 [×2], C2×Dic10, C2×C4×D5, C2×D20 [×2], C20.44D4, D20⋊5C4, C5×C4⋊C8, C4×D20, C4.D20, C2×C40⋊C2, C2×D40, D20.19D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C4○D8, C8⋊C22, D20 [×2], 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)
(1 155)(2 154)(3 153)(4 152)(5 151)(6 150)(7 149)(8 148)(9 147)(10 146)(11 145)(12 144)(13 143)(14 142)(15 141)(16 160)(17 159)(18 158)(19 157)(20 156)(21 138)(22 137)(23 136)(24 135)(25 134)(26 133)(27 132)(28 131)(29 130)(30 129)(31 128)(32 127)(33 126)(34 125)(35 124)(36 123)(37 122)(38 121)(39 140)(40 139)(41 93)(42 92)(43 91)(44 90)(45 89)(46 88)(47 87)(48 86)(49 85)(50 84)(51 83)(52 82)(53 81)(54 100)(55 99)(56 98)(57 97)(58 96)(59 95)(60 94)(61 118)(62 117)(63 116)(64 115)(65 114)(66 113)(67 112)(68 111)(69 110)(70 109)(71 108)(72 107)(73 106)(74 105)(75 104)(76 103)(77 102)(78 101)(79 120)(80 119)
(1 31 47 70)(2 32 48 71)(3 33 49 72)(4 34 50 73)(5 35 51 74)(6 36 52 75)(7 37 53 76)(8 38 54 77)(9 39 55 78)(10 40 56 79)(11 21 57 80)(12 22 58 61)(13 23 59 62)(14 24 60 63)(15 25 41 64)(16 26 42 65)(17 27 43 66)(18 28 44 67)(19 29 45 68)(20 30 46 69)(81 103 149 122)(82 104 150 123)(83 105 151 124)(84 106 152 125)(85 107 153 126)(86 108 154 127)(87 109 155 128)(88 110 156 129)(89 111 157 130)(90 112 158 131)(91 113 159 132)(92 114 160 133)(93 115 141 134)(94 116 142 135)(95 117 143 136)(96 118 144 137)(97 119 145 138)(98 120 146 139)(99 101 147 140)(100 102 148 121)
(1 83 16 98 11 93 6 88)(2 84 17 99 12 94 7 89)(3 85 18 100 13 95 8 90)(4 86 19 81 14 96 9 91)(5 87 20 82 15 97 10 92)(21 124 36 139 31 134 26 129)(22 125 37 140 32 135 27 130)(23 126 38 121 33 136 28 131)(24 127 39 122 34 137 29 132)(25 128 40 123 35 138 30 133)(41 145 56 160 51 155 46 150)(42 146 57 141 52 156 47 151)(43 147 58 142 53 157 48 152)(44 148 59 143 54 158 49 153)(45 149 60 144 55 159 50 154)(61 106 76 101 71 116 66 111)(62 107 77 102 72 117 67 112)(63 108 78 103 73 118 68 113)(64 109 79 104 74 119 69 114)(65 110 80 105 75 120 70 115)
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), (1,155)(2,154)(3,153)(4,152)(5,151)(6,150)(7,149)(8,148)(9,147)(10,146)(11,145)(12,144)(13,143)(14,142)(15,141)(16,160)(17,159)(18,158)(19,157)(20,156)(21,138)(22,137)(23,136)(24,135)(25,134)(26,133)(27,132)(28,131)(29,130)(30,129)(31,128)(32,127)(33,126)(34,125)(35,124)(36,123)(37,122)(38,121)(39,140)(40,139)(41,93)(42,92)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86)(49,85)(50,84)(51,83)(52,82)(53,81)(54,100)(55,99)(56,98)(57,97)(58,96)(59,95)(60,94)(61,118)(62,117)(63,116)(64,115)(65,114)(66,113)(67,112)(68,111)(69,110)(70,109)(71,108)(72,107)(73,106)(74,105)(75,104)(76,103)(77,102)(78,101)(79,120)(80,119), (1,31,47,70)(2,32,48,71)(3,33,49,72)(4,34,50,73)(5,35,51,74)(6,36,52,75)(7,37,53,76)(8,38,54,77)(9,39,55,78)(10,40,56,79)(11,21,57,80)(12,22,58,61)(13,23,59,62)(14,24,60,63)(15,25,41,64)(16,26,42,65)(17,27,43,66)(18,28,44,67)(19,29,45,68)(20,30,46,69)(81,103,149,122)(82,104,150,123)(83,105,151,124)(84,106,152,125)(85,107,153,126)(86,108,154,127)(87,109,155,128)(88,110,156,129)(89,111,157,130)(90,112,158,131)(91,113,159,132)(92,114,160,133)(93,115,141,134)(94,116,142,135)(95,117,143,136)(96,118,144,137)(97,119,145,138)(98,120,146,139)(99,101,147,140)(100,102,148,121), (1,83,16,98,11,93,6,88)(2,84,17,99,12,94,7,89)(3,85,18,100,13,95,8,90)(4,86,19,81,14,96,9,91)(5,87,20,82,15,97,10,92)(21,124,36,139,31,134,26,129)(22,125,37,140,32,135,27,130)(23,126,38,121,33,136,28,131)(24,127,39,122,34,137,29,132)(25,128,40,123,35,138,30,133)(41,145,56,160,51,155,46,150)(42,146,57,141,52,156,47,151)(43,147,58,142,53,157,48,152)(44,148,59,143,54,158,49,153)(45,149,60,144,55,159,50,154)(61,106,76,101,71,116,66,111)(62,107,77,102,72,117,67,112)(63,108,78,103,73,118,68,113)(64,109,79,104,74,119,69,114)(65,110,80,105,75,120,70,115)>;
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), (1,155)(2,154)(3,153)(4,152)(5,151)(6,150)(7,149)(8,148)(9,147)(10,146)(11,145)(12,144)(13,143)(14,142)(15,141)(16,160)(17,159)(18,158)(19,157)(20,156)(21,138)(22,137)(23,136)(24,135)(25,134)(26,133)(27,132)(28,131)(29,130)(30,129)(31,128)(32,127)(33,126)(34,125)(35,124)(36,123)(37,122)(38,121)(39,140)(40,139)(41,93)(42,92)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86)(49,85)(50,84)(51,83)(52,82)(53,81)(54,100)(55,99)(56,98)(57,97)(58,96)(59,95)(60,94)(61,118)(62,117)(63,116)(64,115)(65,114)(66,113)(67,112)(68,111)(69,110)(70,109)(71,108)(72,107)(73,106)(74,105)(75,104)(76,103)(77,102)(78,101)(79,120)(80,119), (1,31,47,70)(2,32,48,71)(3,33,49,72)(4,34,50,73)(5,35,51,74)(6,36,52,75)(7,37,53,76)(8,38,54,77)(9,39,55,78)(10,40,56,79)(11,21,57,80)(12,22,58,61)(13,23,59,62)(14,24,60,63)(15,25,41,64)(16,26,42,65)(17,27,43,66)(18,28,44,67)(19,29,45,68)(20,30,46,69)(81,103,149,122)(82,104,150,123)(83,105,151,124)(84,106,152,125)(85,107,153,126)(86,108,154,127)(87,109,155,128)(88,110,156,129)(89,111,157,130)(90,112,158,131)(91,113,159,132)(92,114,160,133)(93,115,141,134)(94,116,142,135)(95,117,143,136)(96,118,144,137)(97,119,145,138)(98,120,146,139)(99,101,147,140)(100,102,148,121), (1,83,16,98,11,93,6,88)(2,84,17,99,12,94,7,89)(3,85,18,100,13,95,8,90)(4,86,19,81,14,96,9,91)(5,87,20,82,15,97,10,92)(21,124,36,139,31,134,26,129)(22,125,37,140,32,135,27,130)(23,126,38,121,33,136,28,131)(24,127,39,122,34,137,29,132)(25,128,40,123,35,138,30,133)(41,145,56,160,51,155,46,150)(42,146,57,141,52,156,47,151)(43,147,58,142,53,157,48,152)(44,148,59,143,54,158,49,153)(45,149,60,144,55,159,50,154)(61,106,76,101,71,116,66,111)(62,107,77,102,72,117,67,112)(63,108,78,103,73,118,68,113)(64,109,79,104,74,119,69,114)(65,110,80,105,75,120,70,115) );
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)], [(1,155),(2,154),(3,153),(4,152),(5,151),(6,150),(7,149),(8,148),(9,147),(10,146),(11,145),(12,144),(13,143),(14,142),(15,141),(16,160),(17,159),(18,158),(19,157),(20,156),(21,138),(22,137),(23,136),(24,135),(25,134),(26,133),(27,132),(28,131),(29,130),(30,129),(31,128),(32,127),(33,126),(34,125),(35,124),(36,123),(37,122),(38,121),(39,140),(40,139),(41,93),(42,92),(43,91),(44,90),(45,89),(46,88),(47,87),(48,86),(49,85),(50,84),(51,83),(52,82),(53,81),(54,100),(55,99),(56,98),(57,97),(58,96),(59,95),(60,94),(61,118),(62,117),(63,116),(64,115),(65,114),(66,113),(67,112),(68,111),(69,110),(70,109),(71,108),(72,107),(73,106),(74,105),(75,104),(76,103),(77,102),(78,101),(79,120),(80,119)], [(1,31,47,70),(2,32,48,71),(3,33,49,72),(4,34,50,73),(5,35,51,74),(6,36,52,75),(7,37,53,76),(8,38,54,77),(9,39,55,78),(10,40,56,79),(11,21,57,80),(12,22,58,61),(13,23,59,62),(14,24,60,63),(15,25,41,64),(16,26,42,65),(17,27,43,66),(18,28,44,67),(19,29,45,68),(20,30,46,69),(81,103,149,122),(82,104,150,123),(83,105,151,124),(84,106,152,125),(85,107,153,126),(86,108,154,127),(87,109,155,128),(88,110,156,129),(89,111,157,130),(90,112,158,131),(91,113,159,132),(92,114,160,133),(93,115,141,134),(94,116,142,135),(95,117,143,136),(96,118,144,137),(97,119,145,138),(98,120,146,139),(99,101,147,140),(100,102,148,121)], [(1,83,16,98,11,93,6,88),(2,84,17,99,12,94,7,89),(3,85,18,100,13,95,8,90),(4,86,19,81,14,96,9,91),(5,87,20,82,15,97,10,92),(21,124,36,139,31,134,26,129),(22,125,37,140,32,135,27,130),(23,126,38,121,33,136,28,131),(24,127,39,122,34,137,29,132),(25,128,40,123,35,138,30,133),(41,145,56,160,51,155,46,150),(42,146,57,141,52,156,47,151),(43,147,58,142,53,157,48,152),(44,148,59,143,54,158,49,153),(45,149,60,144,55,159,50,154),(61,106,76,101,71,116,66,111),(62,107,77,102,72,117,67,112),(63,108,78,103,73,118,68,113),(64,109,79,104,74,119,69,114),(65,110,80,105,75,120,70,115)])
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