metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.1Q8, C4.4(Q8×D5), C5⋊4(D4.Q8), C4.Q8⋊12D5, C20.16(C2×Q8), C4⋊C4.165D10, (C2×C8).142D10, C4.Dic10⋊6C2, D20⋊8C4.6C2, D20⋊6C4.6C2, C10.58(C4○D8), C4.77(C4○D20), C10.D8⋊18C2, C20.8Q8⋊31C2, (C2×Dic5).53D4, C22.222(D4×D5), D20⋊5C4.14C2, C20.169(C4○D4), C2.25(D40⋊C2), C10.74(C8⋊C22), (C2×C40).289C22, (C2×C20).287C23, (C2×D20).83C22, C10.38(C22⋊Q8), C2.15(D10⋊Q8), C4⋊Dic5.115C22, (C4×Dic5).39C22, C2.25(SD16⋊3D5), (C5×C4.Q8)⋊20C2, (C2×C10).292(C2×D4), (C5×C4⋊C4).80C22, (C2×C5⋊2C8).64C22, (C2×C4).390(C22×D5), SmallGroup(320,498)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.Q8
G = < a,b,c,d | a20=b2=1, c4=a10, d2=a5c2, bab=a-1, ac=ca, dad-1=a11, cbc-1=a15b, bd=db, dcd-1=c3 >
Subgroups: 454 in 102 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C5⋊2C8, C40, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, D4.Q8, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C10.D8, D20⋊6C4, C20.8Q8, D20⋊5C4, C5×C4.Q8, C4.Dic10, D20⋊8C4, D20.Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C4○D8, C8⋊C22, C22×D5, D4.Q8, C4○D20, D4×D5, Q8×D5, D10⋊Q8, D40⋊C2, SD16⋊3D5, D20.Q8
►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)(21 102)(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)(41 68)(42 67)(43 66)(44 65)(45 64)(46 63)(47 62)(48 61)(49 80)(50 79)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)(59 70)(60 69)(81 129)(82 128)(83 127)(84 126)(85 125)(86 124)(87 123)(88 122)(89 121)(90 140)(91 139)(92 138)(93 137)(94 136)(95 135)(96 134)(97 133)(98 132)(99 131)(100 130)
(1 89 146 127 11 99 156 137)(2 90 147 128 12 100 157 138)(3 91 148 129 13 81 158 139)(4 92 149 130 14 82 159 140)(5 93 150 131 15 83 160 121)(6 94 151 132 16 84 141 122)(7 95 152 133 17 85 142 123)(8 96 153 134 18 86 143 124)(9 97 154 135 19 87 144 125)(10 98 155 136 20 88 145 126)(21 78 115 59 31 68 105 49)(22 79 116 60 32 69 106 50)(23 80 117 41 33 70 107 51)(24 61 118 42 34 71 108 52)(25 62 119 43 35 72 109 53)(26 63 120 44 36 73 110 54)(27 64 101 45 37 74 111 55)(28 65 102 46 38 75 112 56)(29 66 103 47 39 76 113 57)(30 67 104 48 40 77 114 58)
(1 40 151 119)(2 31 152 110)(3 22 153 101)(4 33 154 112)(5 24 155 103)(6 35 156 114)(7 26 157 105)(8 37 158 116)(9 28 159 107)(10 39 160 118)(11 30 141 109)(12 21 142 120)(13 32 143 111)(14 23 144 102)(15 34 145 113)(16 25 146 104)(17 36 147 115)(18 27 148 106)(19 38 149 117)(20 29 150 108)(41 125 75 82)(42 136 76 93)(43 127 77 84)(44 138 78 95)(45 129 79 86)(46 140 80 97)(47 131 61 88)(48 122 62 99)(49 133 63 90)(50 124 64 81)(51 135 65 92)(52 126 66 83)(53 137 67 94)(54 128 68 85)(55 139 69 96)(56 130 70 87)(57 121 71 98)(58 132 72 89)(59 123 73 100)(60 134 74 91)
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)(21,102)(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)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(81,129)(82,128)(83,127)(84,126)(85,125)(86,124)(87,123)(88,122)(89,121)(90,140)(91,139)(92,138)(93,137)(94,136)(95,135)(96,134)(97,133)(98,132)(99,131)(100,130), (1,89,146,127,11,99,156,137)(2,90,147,128,12,100,157,138)(3,91,148,129,13,81,158,139)(4,92,149,130,14,82,159,140)(5,93,150,131,15,83,160,121)(6,94,151,132,16,84,141,122)(7,95,152,133,17,85,142,123)(8,96,153,134,18,86,143,124)(9,97,154,135,19,87,144,125)(10,98,155,136,20,88,145,126)(21,78,115,59,31,68,105,49)(22,79,116,60,32,69,106,50)(23,80,117,41,33,70,107,51)(24,61,118,42,34,71,108,52)(25,62,119,43,35,72,109,53)(26,63,120,44,36,73,110,54)(27,64,101,45,37,74,111,55)(28,65,102,46,38,75,112,56)(29,66,103,47,39,76,113,57)(30,67,104,48,40,77,114,58), (1,40,151,119)(2,31,152,110)(3,22,153,101)(4,33,154,112)(5,24,155,103)(6,35,156,114)(7,26,157,105)(8,37,158,116)(9,28,159,107)(10,39,160,118)(11,30,141,109)(12,21,142,120)(13,32,143,111)(14,23,144,102)(15,34,145,113)(16,25,146,104)(17,36,147,115)(18,27,148,106)(19,38,149,117)(20,29,150,108)(41,125,75,82)(42,136,76,93)(43,127,77,84)(44,138,78,95)(45,129,79,86)(46,140,80,97)(47,131,61,88)(48,122,62,99)(49,133,63,90)(50,124,64,81)(51,135,65,92)(52,126,66,83)(53,137,67,94)(54,128,68,85)(55,139,69,96)(56,130,70,87)(57,121,71,98)(58,132,72,89)(59,123,73,100)(60,134,74,91)>;
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)(21,102)(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)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,80)(50,79)(51,78)(52,77)(53,76)(54,75)(55,74)(56,73)(57,72)(58,71)(59,70)(60,69)(81,129)(82,128)(83,127)(84,126)(85,125)(86,124)(87,123)(88,122)(89,121)(90,140)(91,139)(92,138)(93,137)(94,136)(95,135)(96,134)(97,133)(98,132)(99,131)(100,130), (1,89,146,127,11,99,156,137)(2,90,147,128,12,100,157,138)(3,91,148,129,13,81,158,139)(4,92,149,130,14,82,159,140)(5,93,150,131,15,83,160,121)(6,94,151,132,16,84,141,122)(7,95,152,133,17,85,142,123)(8,96,153,134,18,86,143,124)(9,97,154,135,19,87,144,125)(10,98,155,136,20,88,145,126)(21,78,115,59,31,68,105,49)(22,79,116,60,32,69,106,50)(23,80,117,41,33,70,107,51)(24,61,118,42,34,71,108,52)(25,62,119,43,35,72,109,53)(26,63,120,44,36,73,110,54)(27,64,101,45,37,74,111,55)(28,65,102,46,38,75,112,56)(29,66,103,47,39,76,113,57)(30,67,104,48,40,77,114,58), (1,40,151,119)(2,31,152,110)(3,22,153,101)(4,33,154,112)(5,24,155,103)(6,35,156,114)(7,26,157,105)(8,37,158,116)(9,28,159,107)(10,39,160,118)(11,30,141,109)(12,21,142,120)(13,32,143,111)(14,23,144,102)(15,34,145,113)(16,25,146,104)(17,36,147,115)(18,27,148,106)(19,38,149,117)(20,29,150,108)(41,125,75,82)(42,136,76,93)(43,127,77,84)(44,138,78,95)(45,129,79,86)(46,140,80,97)(47,131,61,88)(48,122,62,99)(49,133,63,90)(50,124,64,81)(51,135,65,92)(52,126,66,83)(53,137,67,94)(54,128,68,85)(55,139,69,96)(56,130,70,87)(57,121,71,98)(58,132,72,89)(59,123,73,100)(60,134,74,91) );
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),(21,102),(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),(41,68),(42,67),(43,66),(44,65),(45,64),(46,63),(47,62),(48,61),(49,80),(50,79),(51,78),(52,77),(53,76),(54,75),(55,74),(56,73),(57,72),(58,71),(59,70),(60,69),(81,129),(82,128),(83,127),(84,126),(85,125),(86,124),(87,123),(88,122),(89,121),(90,140),(91,139),(92,138),(93,137),(94,136),(95,135),(96,134),(97,133),(98,132),(99,131),(100,130)], [(1,89,146,127,11,99,156,137),(2,90,147,128,12,100,157,138),(3,91,148,129,13,81,158,139),(4,92,149,130,14,82,159,140),(5,93,150,131,15,83,160,121),(6,94,151,132,16,84,141,122),(7,95,152,133,17,85,142,123),(8,96,153,134,18,86,143,124),(9,97,154,135,19,87,144,125),(10,98,155,136,20,88,145,126),(21,78,115,59,31,68,105,49),(22,79,116,60,32,69,106,50),(23,80,117,41,33,70,107,51),(24,61,118,42,34,71,108,52),(25,62,119,43,35,72,109,53),(26,63,120,44,36,73,110,54),(27,64,101,45,37,74,111,55),(28,65,102,46,38,75,112,56),(29,66,103,47,39,76,113,57),(30,67,104,48,40,77,114,58)], [(1,40,151,119),(2,31,152,110),(3,22,153,101),(4,33,154,112),(5,24,155,103),(6,35,156,114),(7,26,157,105),(8,37,158,116),(9,28,159,107),(10,39,160,118),(11,30,141,109),(12,21,142,120),(13,32,143,111),(14,23,144,102),(15,34,145,113),(16,25,146,104),(17,36,147,115),(18,27,148,106),(19,38,149,117),(20,29,150,108),(41,125,75,82),(42,136,76,93),(43,127,77,84),(44,138,78,95),(45,129,79,86),(46,140,80,97),(47,131,61,88),(48,122,62,99),(49,133,63,90),(50,124,64,81),(51,135,65,92),(52,126,66,83),(53,137,67,94),(54,128,68,85),(55,139,69,96),(56,130,70,87),(57,121,71,98),(58,132,72,89),(59,123,73,100),(60,134,74,91)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 10 | 10 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | Q8 | D4 | D5 | C4○D4 | D10 | D10 | C4○D8 | C4○D20 | C8⋊C22 | Q8×D5 | D4×D5 | D40⋊C2 | SD16⋊3D5 |
| kernel | D20.Q8 | C10.D8 | D20⋊6C4 | C20.8Q8 | D20⋊5C4 | C5×C4.Q8 | C4.Dic10 | D20⋊8C4 | D20 | C2×Dic5 | C4.Q8 | C20 | C4⋊C4 | C2×C8 | C10 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D20.Q8 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 32 |
| 0 | 0 | 0 | 0 | 40 | 20 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 9 | 1 |
| 1 | 39 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 12 |
| 0 | 0 | 0 | 0 | 15 | 2 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 1 | 32 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,21,40,0,0,0,0,32,20],[40,40,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,9,0,0,0,0,0,1],[1,0,0,0,0,0,39,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,28,15,0,0,0,0,12,2],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,1,0,0,0,0,0,32] >;
D20.Q8 in GAP, Magma, Sage, TeX
D_{20}.Q_8 % in TeX
G:=Group("D20.Q8"); // GroupNames label
G:=SmallGroup(320,498);
// by ID
G=gap.SmallGroup(320,498);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,64,590,555,268,1684,851,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=1,c^4=a^10,d^2=a^5*c^2,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations