metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.5D20, C40.84D4, Q8.5D20, M4(2).35D10, C8○D4.1D5, (C5×D4).22D4, C4.21(C2×D20), C20.44(C2×D4), (C2×C8).82D10, (C5×Q8).22D4, C4○D4.33D10, C5⋊3(D4.5D4), C8.41(C5⋊D4), C40.6C4⋊16C2, (C2×Dic20)⋊13C2, (C2×C40).68C22, C4.12D20⋊14C2, C10.78(C4⋊D4), C2.26(C20⋊7D4), (C2×C20).423C23, D4.9D10.1C2, C22.10(C4○D20), C4.Dic5.18C22, (C5×M4(2)).38C22, (C2×Dic10).122C22, (C5×C8○D4).1C2, C4.119(C2×C5⋊D4), (C2×C10).8(C4○D4), (C5×C4○D4).38C22, (C2×C4).125(C22×D5), SmallGroup(320,770)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.5D20
G = < a,b,c,d | a4=b2=1, c20=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2c19 >
Subgroups: 350 in 100 conjugacy classes, 39 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C10, C10, C2×C8, C2×C8, M4(2), M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, C5⋊2C8, C40, C40, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, D4.5D4, Dic20, C4.Dic5, D4.D5, C5⋊Q16, C2×C40, C2×C40, C5×M4(2), C5×M4(2), C2×Dic10, C5×C4○D4, C40.6C4, C4.12D20, C2×Dic20, D4.9D10, C5×C8○D4, D4.5D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, D20, C5⋊D4, C22×D5, D4.5D4, C2×D20, C4○D20, C2×C5⋊D4, C20⋊7D4, D4.5D20
►On 160 points
Generators in S160
(1 121 21 141)(2 122 22 142)(3 123 23 143)(4 124 24 144)(5 125 25 145)(6 126 26 146)(7 127 27 147)(8 128 28 148)(9 129 29 149)(10 130 30 150)(11 131 31 151)(12 132 32 152)(13 133 33 153)(14 134 34 154)(15 135 35 155)(16 136 36 156)(17 137 37 157)(18 138 38 158)(19 139 39 159)(20 140 40 160)(41 88 61 108)(42 89 62 109)(43 90 63 110)(44 91 64 111)(45 92 65 112)(46 93 66 113)(47 94 67 114)(48 95 68 115)(49 96 69 116)(50 97 70 117)(51 98 71 118)(52 99 72 119)(53 100 73 120)(54 101 74 81)(55 102 75 82)(56 103 76 83)(57 104 77 84)(58 105 78 85)(59 106 79 86)(60 107 80 87)
(1 141)(2 142)(3 143)(4 144)(5 145)(6 146)(7 147)(8 148)(9 149)(10 150)(11 151)(12 152)(13 153)(14 154)(15 155)(16 156)(17 157)(18 158)(19 159)(20 160)(21 121)(22 122)(23 123)(24 124)(25 125)(26 126)(27 127)(28 128)(29 129)(30 130)(31 131)(32 132)(33 133)(34 134)(35 135)(36 136)(37 137)(38 138)(39 139)(40 140)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)
(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 59 21 79)(2 58 22 78)(3 57 23 77)(4 56 24 76)(5 55 25 75)(6 54 26 74)(7 53 27 73)(8 52 28 72)(9 51 29 71)(10 50 30 70)(11 49 31 69)(12 48 32 68)(13 47 33 67)(14 46 34 66)(15 45 35 65)(16 44 36 64)(17 43 37 63)(18 42 38 62)(19 41 39 61)(20 80 40 60)(81 146 101 126)(82 145 102 125)(83 144 103 124)(84 143 104 123)(85 142 105 122)(86 141 106 121)(87 140 107 160)(88 139 108 159)(89 138 109 158)(90 137 110 157)(91 136 111 156)(92 135 112 155)(93 134 113 154)(94 133 114 153)(95 132 115 152)(96 131 116 151)(97 130 117 150)(98 129 118 149)(99 128 119 148)(100 127 120 147)
G:=sub<Sym(160)| (1,121,21,141)(2,122,22,142)(3,123,23,143)(4,124,24,144)(5,125,25,145)(6,126,26,146)(7,127,27,147)(8,128,28,148)(9,129,29,149)(10,130,30,150)(11,131,31,151)(12,132,32,152)(13,133,33,153)(14,134,34,154)(15,135,35,155)(16,136,36,156)(17,137,37,157)(18,138,38,158)(19,139,39,159)(20,140,40,160)(41,88,61,108)(42,89,62,109)(43,90,63,110)(44,91,64,111)(45,92,65,112)(46,93,66,113)(47,94,67,114)(48,95,68,115)(49,96,69,116)(50,97,70,117)(51,98,71,118)(52,99,72,119)(53,100,73,120)(54,101,74,81)(55,102,75,82)(56,103,76,83)(57,104,77,84)(58,105,78,85)(59,106,79,86)(60,107,80,87), (1,141)(2,142)(3,143)(4,144)(5,145)(6,146)(7,147)(8,148)(9,149)(10,150)(11,151)(12,152)(13,153)(14,154)(15,155)(16,156)(17,157)(18,158)(19,159)(20,160)(21,121)(22,122)(23,123)(24,124)(25,125)(26,126)(27,127)(28,128)(29,129)(30,130)(31,131)(32,132)(33,133)(34,134)(35,135)(36,136)(37,137)(38,138)(39,139)(40,140)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (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), (1,59,21,79)(2,58,22,78)(3,57,23,77)(4,56,24,76)(5,55,25,75)(6,54,26,74)(7,53,27,73)(8,52,28,72)(9,51,29,71)(10,50,30,70)(11,49,31,69)(12,48,32,68)(13,47,33,67)(14,46,34,66)(15,45,35,65)(16,44,36,64)(17,43,37,63)(18,42,38,62)(19,41,39,61)(20,80,40,60)(81,146,101,126)(82,145,102,125)(83,144,103,124)(84,143,104,123)(85,142,105,122)(86,141,106,121)(87,140,107,160)(88,139,108,159)(89,138,109,158)(90,137,110,157)(91,136,111,156)(92,135,112,155)(93,134,113,154)(94,133,114,153)(95,132,115,152)(96,131,116,151)(97,130,117,150)(98,129,118,149)(99,128,119,148)(100,127,120,147)>;
G:=Group( (1,121,21,141)(2,122,22,142)(3,123,23,143)(4,124,24,144)(5,125,25,145)(6,126,26,146)(7,127,27,147)(8,128,28,148)(9,129,29,149)(10,130,30,150)(11,131,31,151)(12,132,32,152)(13,133,33,153)(14,134,34,154)(15,135,35,155)(16,136,36,156)(17,137,37,157)(18,138,38,158)(19,139,39,159)(20,140,40,160)(41,88,61,108)(42,89,62,109)(43,90,63,110)(44,91,64,111)(45,92,65,112)(46,93,66,113)(47,94,67,114)(48,95,68,115)(49,96,69,116)(50,97,70,117)(51,98,71,118)(52,99,72,119)(53,100,73,120)(54,101,74,81)(55,102,75,82)(56,103,76,83)(57,104,77,84)(58,105,78,85)(59,106,79,86)(60,107,80,87), (1,141)(2,142)(3,143)(4,144)(5,145)(6,146)(7,147)(8,148)(9,149)(10,150)(11,151)(12,152)(13,153)(14,154)(15,155)(16,156)(17,157)(18,158)(19,159)(20,160)(21,121)(22,122)(23,123)(24,124)(25,125)(26,126)(27,127)(28,128)(29,129)(30,130)(31,131)(32,132)(33,133)(34,134)(35,135)(36,136)(37,137)(38,138)(39,139)(40,140)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (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), (1,59,21,79)(2,58,22,78)(3,57,23,77)(4,56,24,76)(5,55,25,75)(6,54,26,74)(7,53,27,73)(8,52,28,72)(9,51,29,71)(10,50,30,70)(11,49,31,69)(12,48,32,68)(13,47,33,67)(14,46,34,66)(15,45,35,65)(16,44,36,64)(17,43,37,63)(18,42,38,62)(19,41,39,61)(20,80,40,60)(81,146,101,126)(82,145,102,125)(83,144,103,124)(84,143,104,123)(85,142,105,122)(86,141,106,121)(87,140,107,160)(88,139,108,159)(89,138,109,158)(90,137,110,157)(91,136,111,156)(92,135,112,155)(93,134,113,154)(94,133,114,153)(95,132,115,152)(96,131,116,151)(97,130,117,150)(98,129,118,149)(99,128,119,148)(100,127,120,147) );
G=PermutationGroup([[(1,121,21,141),(2,122,22,142),(3,123,23,143),(4,124,24,144),(5,125,25,145),(6,126,26,146),(7,127,27,147),(8,128,28,148),(9,129,29,149),(10,130,30,150),(11,131,31,151),(12,132,32,152),(13,133,33,153),(14,134,34,154),(15,135,35,155),(16,136,36,156),(17,137,37,157),(18,138,38,158),(19,139,39,159),(20,140,40,160),(41,88,61,108),(42,89,62,109),(43,90,63,110),(44,91,64,111),(45,92,65,112),(46,93,66,113),(47,94,67,114),(48,95,68,115),(49,96,69,116),(50,97,70,117),(51,98,71,118),(52,99,72,119),(53,100,73,120),(54,101,74,81),(55,102,75,82),(56,103,76,83),(57,104,77,84),(58,105,78,85),(59,106,79,86),(60,107,80,87)], [(1,141),(2,142),(3,143),(4,144),(5,145),(6,146),(7,147),(8,148),(9,149),(10,150),(11,151),(12,152),(13,153),(14,154),(15,155),(16,156),(17,157),(18,158),(19,159),(20,160),(21,121),(22,122),(23,123),(24,124),(25,125),(26,126),(27,127),(28,128),(29,129),(30,130),(31,131),(32,132),(33,133),(34,134),(35,135),(36,136),(37,137),(38,138),(39,139),(40,140),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80)], [(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)], [(1,59,21,79),(2,58,22,78),(3,57,23,77),(4,56,24,76),(5,55,25,75),(6,54,26,74),(7,53,27,73),(8,52,28,72),(9,51,29,71),(10,50,30,70),(11,49,31,69),(12,48,32,68),(13,47,33,67),(14,46,34,66),(15,45,35,65),(16,44,36,64),(17,43,37,63),(18,42,38,62),(19,41,39,61),(20,80,40,60),(81,146,101,126),(82,145,102,125),(83,144,103,124),(84,143,104,123),(85,142,105,122),(86,141,106,121),(87,140,107,160),(88,139,108,159),(89,138,109,158),(90,137,110,157),(91,136,111,156),(92,135,112,155),(93,134,113,154),(94,133,114,153),(95,132,115,152),(96,131,116,151),(97,130,117,150),(98,129,118,149),(99,128,119,148),(100,127,120,147)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 10A | 10B | 10C | ··· | 10H | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 40A | ··· | 40H | 40I | ··· | 40T |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 40 | 40 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 40 | 40 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | D20 | D20 | C4○D20 | D4.5D4 | D4.5D20 |
| kernel | D4.5D20 | C40.6C4 | C4.12D20 | C2×Dic20 | D4.9D10 | C5×C8○D4 | C40 | C5×D4 | C5×Q8 | C8○D4 | C2×C10 | C2×C8 | M4(2) | C4○D4 | C8 | D4 | Q8 | C22 | C5 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 8 | 2 | 8 |
Matrix representation of D4.5D20 ►in GL4(𝔽41) generated by
| 1 | 0 | 15 | 17 |
| 5 | 40 | 39 | 0 |
| 0 | 1 | 18 | 22 |
| 24 | 16 | 17 | 23 |
| 1 | 0 | 15 | 17 |
| 5 | 40 | 39 | 0 |
| 0 | 0 | 18 | 22 |
| 0 | 0 | 17 | 23 |
| 13 | 0 | 0 | 0 |
| 10 | 19 | 0 | 0 |
| 16 | 0 | 26 | 38 |
| 10 | 0 | 3 | 6 |
| 25 | 31 | 33 | 5 |
| 13 | 3 | 5 | 5 |
| 7 | 35 | 16 | 28 |
| 35 | 30 | 10 | 38 |
G:=sub<GL(4,GF(41))| [1,5,0,24,0,40,1,16,15,39,18,17,17,0,22,23],[1,5,0,0,0,40,0,0,15,39,18,17,17,0,22,23],[13,10,16,10,0,19,0,0,0,0,26,3,0,0,38,6],[25,13,7,35,31,3,35,30,33,5,16,10,5,5,28,38] >;
D4.5D20 in GAP, Magma, Sage, TeX
D_4._5D_{20} % in TeX
G:=Group("D4.5D20"); // GroupNames label
G:=SmallGroup(320,770);
// by ID
G=gap.SmallGroup(320,770);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,344,254,1123,297,136,1684,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^20=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=a^2*c^19>;
// generators/relations