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