metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.47D4, D8.10D10, C40.39C23, C20.18C24, Q16.12D10, SD16.2D10, Dic10.47D4, Dic10.12C23, Dic20.16C22, C4○D8⋊6D5, C5⋊3(Q8○D8), (D5×Q16)⋊7C2, C5⋊D4.3D4, D8⋊3D5⋊7C2, C4.145(D4×D5), D4.D5.C22, C4○D4.13D10, D10.54(C2×D4), (C2×C8).106D10, SD16⋊D5⋊6C2, C20.351(C2×D4), C5⋊2C8.9C23, (C8×D5).8C22, C4.18(C23×D5), C22.10(D4×D5), C8.18(C22×D5), (C2×Dic20)⋊23C2, D4.9D10⋊8C2, D20.3C4⋊8C2, (Q8×D5).2C22, Dic5.60(C2×D4), (C5×D4).12C23, D4.12(C22×D5), (C4×D5).11C23, (C5×D8).10C22, C8⋊D5.2C22, D4.10D10⋊6C2, Q8.12(C22×D5), (C5×Q8).12C23, C5⋊Q16.2C22, (C2×C20).535C23, (C2×C40).106C22, C4○D20.56C22, D4⋊2D5.2C22, C10.119(C22×D4), (C5×Q16).12C22, (C5×SD16).2C22, C4.Dic5.49C22, (C2×Dic10).206C22, C2.92(C2×D4×D5), (C5×C4○D8)⋊6C2, (C2×C10).15(C2×D4), (C5×C4○D4).23C22, (C2×C4).234(C22×D5), SmallGroup(320,1443)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.47D4
G = < a,b,c,d | a8=b2=c10=1, d2=a4, bab=a-1, ac=ca, ad=da, cbc-1=a4b, bd=db, dcd-1=a4c-1 >
Subgroups: 854 in 248 conjugacy classes, 99 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, D5, C10, C10, C2×C8, C2×C8, M4(2), D8, SD16, SD16, Q16, Q16, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C8○D4, C2×Q16, C4○D8, C4○D8, C8.C22, 2- 1+4, C5⋊2C8, C40, Dic10, Dic10, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C5⋊D4, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, Q8○D8, C8×D5, C8⋊D5, Dic20, C4.Dic5, D4.D5, C5⋊Q16, C2×C40, C5×D8, C5×SD16, C5×Q16, C2×Dic10, C2×Dic10, C4○D20, C4○D20, D4⋊2D5, D4⋊2D5, Q8×D5, C5×C4○D4, D20.3C4, C2×Dic20, D8⋊3D5, SD16⋊D5, D5×Q16, D4.9D10, C5×C4○D8, D4.10D10, D20.47D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, C22×D5, Q8○D8, D4×D5, C23×D5, C2×D4×D5, D20.47D4
►On 160 points
Generators in S160
(1 17 22 153 123 86 60 95)(2 18 23 154 124 87 51 96)(3 19 24 155 125 88 52 97)(4 20 25 156 126 89 53 98)(5 11 26 157 127 90 54 99)(6 12 27 158 128 81 55 100)(7 13 28 159 129 82 56 91)(8 14 29 160 130 83 57 92)(9 15 30 151 121 84 58 93)(10 16 21 152 122 85 59 94)(31 142 104 41 69 114 132 79)(32 143 105 42 70 115 133 80)(33 144 106 43 61 116 134 71)(34 145 107 44 62 117 135 72)(35 146 108 45 63 118 136 73)(36 147 109 46 64 119 137 74)(37 148 110 47 65 120 138 75)(38 149 101 48 66 111 139 76)(39 150 102 49 67 112 140 77)(40 141 103 50 68 113 131 78)
(1 148)(2 111)(3 150)(4 113)(5 142)(6 115)(7 144)(8 117)(9 146)(10 119)(11 31)(12 70)(13 33)(14 62)(15 35)(16 64)(17 37)(18 66)(19 39)(20 68)(21 46)(22 75)(23 48)(24 77)(25 50)(26 79)(27 42)(28 71)(29 44)(30 73)(32 81)(34 83)(36 85)(38 87)(40 89)(41 54)(43 56)(45 58)(47 60)(49 52)(51 76)(53 78)(55 80)(57 72)(59 74)(61 82)(63 84)(65 86)(67 88)(69 90)(91 106)(92 135)(93 108)(94 137)(95 110)(96 139)(97 102)(98 131)(99 104)(100 133)(101 154)(103 156)(105 158)(107 160)(109 152)(112 125)(114 127)(116 129)(118 121)(120 123)(122 147)(124 149)(126 141)(128 143)(130 145)(132 157)(134 159)(136 151)(138 153)(140 155)
(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 10 123 122)(2 121 124 9)(3 8 125 130)(4 129 126 7)(5 6 127 128)(11 12 90 81)(13 20 82 89)(14 88 83 19)(15 18 84 87)(16 86 85 17)(21 60 59 22)(23 58 51 30)(24 29 52 57)(25 56 53 28)(26 27 54 55)(31 70 69 32)(33 68 61 40)(34 39 62 67)(35 66 63 38)(36 37 64 65)(41 80 79 42)(43 78 71 50)(44 49 72 77)(45 76 73 48)(46 47 74 75)(91 98 159 156)(92 155 160 97)(93 96 151 154)(94 153 152 95)(99 100 157 158)(101 108 139 136)(102 135 140 107)(103 106 131 134)(104 133 132 105)(109 110 137 138)(111 118 149 146)(112 145 150 117)(113 116 141 144)(114 143 142 115)(119 120 147 148)
G:=sub<Sym(160)| (1,17,22,153,123,86,60,95)(2,18,23,154,124,87,51,96)(3,19,24,155,125,88,52,97)(4,20,25,156,126,89,53,98)(5,11,26,157,127,90,54,99)(6,12,27,158,128,81,55,100)(7,13,28,159,129,82,56,91)(8,14,29,160,130,83,57,92)(9,15,30,151,121,84,58,93)(10,16,21,152,122,85,59,94)(31,142,104,41,69,114,132,79)(32,143,105,42,70,115,133,80)(33,144,106,43,61,116,134,71)(34,145,107,44,62,117,135,72)(35,146,108,45,63,118,136,73)(36,147,109,46,64,119,137,74)(37,148,110,47,65,120,138,75)(38,149,101,48,66,111,139,76)(39,150,102,49,67,112,140,77)(40,141,103,50,68,113,131,78), (1,148)(2,111)(3,150)(4,113)(5,142)(6,115)(7,144)(8,117)(9,146)(10,119)(11,31)(12,70)(13,33)(14,62)(15,35)(16,64)(17,37)(18,66)(19,39)(20,68)(21,46)(22,75)(23,48)(24,77)(25,50)(26,79)(27,42)(28,71)(29,44)(30,73)(32,81)(34,83)(36,85)(38,87)(40,89)(41,54)(43,56)(45,58)(47,60)(49,52)(51,76)(53,78)(55,80)(57,72)(59,74)(61,82)(63,84)(65,86)(67,88)(69,90)(91,106)(92,135)(93,108)(94,137)(95,110)(96,139)(97,102)(98,131)(99,104)(100,133)(101,154)(103,156)(105,158)(107,160)(109,152)(112,125)(114,127)(116,129)(118,121)(120,123)(122,147)(124,149)(126,141)(128,143)(130,145)(132,157)(134,159)(136,151)(138,153)(140,155), (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), (1,10,123,122)(2,121,124,9)(3,8,125,130)(4,129,126,7)(5,6,127,128)(11,12,90,81)(13,20,82,89)(14,88,83,19)(15,18,84,87)(16,86,85,17)(21,60,59,22)(23,58,51,30)(24,29,52,57)(25,56,53,28)(26,27,54,55)(31,70,69,32)(33,68,61,40)(34,39,62,67)(35,66,63,38)(36,37,64,65)(41,80,79,42)(43,78,71,50)(44,49,72,77)(45,76,73,48)(46,47,74,75)(91,98,159,156)(92,155,160,97)(93,96,151,154)(94,153,152,95)(99,100,157,158)(101,108,139,136)(102,135,140,107)(103,106,131,134)(104,133,132,105)(109,110,137,138)(111,118,149,146)(112,145,150,117)(113,116,141,144)(114,143,142,115)(119,120,147,148)>;
G:=Group( (1,17,22,153,123,86,60,95)(2,18,23,154,124,87,51,96)(3,19,24,155,125,88,52,97)(4,20,25,156,126,89,53,98)(5,11,26,157,127,90,54,99)(6,12,27,158,128,81,55,100)(7,13,28,159,129,82,56,91)(8,14,29,160,130,83,57,92)(9,15,30,151,121,84,58,93)(10,16,21,152,122,85,59,94)(31,142,104,41,69,114,132,79)(32,143,105,42,70,115,133,80)(33,144,106,43,61,116,134,71)(34,145,107,44,62,117,135,72)(35,146,108,45,63,118,136,73)(36,147,109,46,64,119,137,74)(37,148,110,47,65,120,138,75)(38,149,101,48,66,111,139,76)(39,150,102,49,67,112,140,77)(40,141,103,50,68,113,131,78), (1,148)(2,111)(3,150)(4,113)(5,142)(6,115)(7,144)(8,117)(9,146)(10,119)(11,31)(12,70)(13,33)(14,62)(15,35)(16,64)(17,37)(18,66)(19,39)(20,68)(21,46)(22,75)(23,48)(24,77)(25,50)(26,79)(27,42)(28,71)(29,44)(30,73)(32,81)(34,83)(36,85)(38,87)(40,89)(41,54)(43,56)(45,58)(47,60)(49,52)(51,76)(53,78)(55,80)(57,72)(59,74)(61,82)(63,84)(65,86)(67,88)(69,90)(91,106)(92,135)(93,108)(94,137)(95,110)(96,139)(97,102)(98,131)(99,104)(100,133)(101,154)(103,156)(105,158)(107,160)(109,152)(112,125)(114,127)(116,129)(118,121)(120,123)(122,147)(124,149)(126,141)(128,143)(130,145)(132,157)(134,159)(136,151)(138,153)(140,155), (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), (1,10,123,122)(2,121,124,9)(3,8,125,130)(4,129,126,7)(5,6,127,128)(11,12,90,81)(13,20,82,89)(14,88,83,19)(15,18,84,87)(16,86,85,17)(21,60,59,22)(23,58,51,30)(24,29,52,57)(25,56,53,28)(26,27,54,55)(31,70,69,32)(33,68,61,40)(34,39,62,67)(35,66,63,38)(36,37,64,65)(41,80,79,42)(43,78,71,50)(44,49,72,77)(45,76,73,48)(46,47,74,75)(91,98,159,156)(92,155,160,97)(93,96,151,154)(94,153,152,95)(99,100,157,158)(101,108,139,136)(102,135,140,107)(103,106,131,134)(104,133,132,105)(109,110,137,138)(111,118,149,146)(112,145,150,117)(113,116,141,144)(114,143,142,115)(119,120,147,148) );
G=PermutationGroup([[(1,17,22,153,123,86,60,95),(2,18,23,154,124,87,51,96),(3,19,24,155,125,88,52,97),(4,20,25,156,126,89,53,98),(5,11,26,157,127,90,54,99),(6,12,27,158,128,81,55,100),(7,13,28,159,129,82,56,91),(8,14,29,160,130,83,57,92),(9,15,30,151,121,84,58,93),(10,16,21,152,122,85,59,94),(31,142,104,41,69,114,132,79),(32,143,105,42,70,115,133,80),(33,144,106,43,61,116,134,71),(34,145,107,44,62,117,135,72),(35,146,108,45,63,118,136,73),(36,147,109,46,64,119,137,74),(37,148,110,47,65,120,138,75),(38,149,101,48,66,111,139,76),(39,150,102,49,67,112,140,77),(40,141,103,50,68,113,131,78)], [(1,148),(2,111),(3,150),(4,113),(5,142),(6,115),(7,144),(8,117),(9,146),(10,119),(11,31),(12,70),(13,33),(14,62),(15,35),(16,64),(17,37),(18,66),(19,39),(20,68),(21,46),(22,75),(23,48),(24,77),(25,50),(26,79),(27,42),(28,71),(29,44),(30,73),(32,81),(34,83),(36,85),(38,87),(40,89),(41,54),(43,56),(45,58),(47,60),(49,52),(51,76),(53,78),(55,80),(57,72),(59,74),(61,82),(63,84),(65,86),(67,88),(69,90),(91,106),(92,135),(93,108),(94,137),(95,110),(96,139),(97,102),(98,131),(99,104),(100,133),(101,154),(103,156),(105,158),(107,160),(109,152),(112,125),(114,127),(116,129),(118,121),(120,123),(122,147),(124,149),(126,141),(128,143),(130,145),(132,157),(134,159),(136,151),(138,153),(140,155)], [(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)], [(1,10,123,122),(2,121,124,9),(3,8,125,130),(4,129,126,7),(5,6,127,128),(11,12,90,81),(13,20,82,89),(14,88,83,19),(15,18,84,87),(16,86,85,17),(21,60,59,22),(23,58,51,30),(24,29,52,57),(25,56,53,28),(26,27,54,55),(31,70,69,32),(33,68,61,40),(34,39,62,67),(35,66,63,38),(36,37,64,65),(41,80,79,42),(43,78,71,50),(44,49,72,77),(45,76,73,48),(46,47,74,75),(91,98,159,156),(92,155,160,97),(93,96,151,154),(94,153,152,95),(99,100,157,158),(101,108,139,136),(102,135,140,107),(103,106,131,134),(104,133,132,105),(109,110,137,138),(111,118,149,146),(112,145,150,117),(113,116,141,144),(114,143,142,115),(119,120,147,148)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 10 | 10 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 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 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | D10 | Q8○D8 | D4×D5 | D4×D5 | D20.47D4 |
| kernel | D20.47D4 | D20.3C4 | C2×Dic20 | D8⋊3D5 | SD16⋊D5 | D5×Q16 | D4.9D10 | C5×C4○D8 | D4.10D10 | Dic10 | D20 | C5⋊D4 | C4○D8 | C2×C8 | D8 | SD16 | Q16 | C4○D4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of D20.47D4 ►in GL4(𝔽41) generated by
| 0 | 0 | 33 | 8 |
| 0 | 0 | 33 | 0 |
| 0 | 36 | 24 | 0 |
| 5 | 36 | 0 | 24 |
| 5 | 29 | 31 | 28 |
| 12 | 36 | 10 | 18 |
| 6 | 18 | 34 | 12 |
| 24 | 24 | 27 | 7 |
| 39 | 16 | 23 | 15 |
| 25 | 25 | 6 | 38 |
| 7 | 7 | 32 | 25 |
| 1 | 14 | 5 | 27 |
| 39 | 13 | 21 | 15 |
| 25 | 2 | 18 | 38 |
| 7 | 35 | 14 | 25 |
| 1 | 33 | 25 | 27 |
G:=sub<GL(4,GF(41))| [0,0,0,5,0,0,36,36,33,33,24,0,8,0,0,24],[5,12,6,24,29,36,18,24,31,10,34,27,28,18,12,7],[39,25,7,1,16,25,7,14,23,6,32,5,15,38,25,27],[39,25,7,1,13,2,35,33,21,18,14,25,15,38,25,27] >;
D20.47D4 in GAP, Magma, Sage, TeX
D_{20}._{47}D_4 % in TeX
G:=Group("D20.47D4"); // GroupNames label
G:=SmallGroup(320,1443);
// by ID
G=gap.SmallGroup(320,1443);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,387,184,570,185,136,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^10=1,d^2=a^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=a^4*c^-1>;
// generators/relations