metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20.23D4, C42.63D10, C4.50(D4×D5), (C4×D20)⋊21C2, C20⋊3C8⋊29C2, C4.4D4⋊1D5, C20.24(C2×D4), (C2×D4).47D10, (C2×C20).271D4, C5⋊5(D4.2D4), (C2×Q8).37D10, C20.68(C4○D4), D4⋊Dic5⋊19C2, Q8⋊Dic5⋊22C2, C4.2(D4⋊2D5), C10.105(C4○D8), C2.11(C20⋊2D4), (C4×C20).106C22, (C2×C20).375C23, (D4×C10).63C22, (Q8×C10).55C22, C2.18(D4⋊D10), C10.102(C4⋊D4), C10.119(C8⋊C22), (C2×D20).252C22, C4⋊Dic5.341C22, C2.24(D4.8D10), (C2×Q8⋊D5)⋊12C2, (C2×D4⋊D5).6C2, (C5×C4.4D4)⋊1C2, (C2×C10).506(C2×D4), (C2×C4).61(C5⋊D4), (C2×C4).475(C22×D5), C22.181(C2×C5⋊D4), (C2×C5⋊2C8).121C22, SmallGroup(320,684)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20.23D4
G = < a,b,c,d | a20=b2=c4=1, d2=a10, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a15b, dcd-1=a10c-1 >
Subgroups: 518 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 [×2], C10 [×3], C10, C42, C22⋊C4 [×3], C4⋊C4, C2×C8 [×2], D8 [×2], SD16 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, Dic5, C20 [×2], C20 [×3], D10 [×4], C2×C10, C2×C10 [×3], D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C5⋊2C8 [×2], C4×D5 [×2], D20 [×2], D20, C2×Dic5, C2×C20 [×3], C2×C20, C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×C10, D4.2D4, C2×C5⋊2C8 [×2], C4⋊Dic5, D10⋊C4, D4⋊D5 [×2], Q8⋊D5 [×2], C4×C20, C5×C22⋊C4 [×2], C2×C4×D5, C2×D20, D4×C10, Q8×C10, C20⋊3C8, D4⋊Dic5, Q8⋊Dic5, C4×D20, C2×D4⋊D5, C2×Q8⋊D5, C5×C4.4D4, D20.23D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C4○D8, C8⋊C22, C5⋊D4 [×2], C22×D5, D4.2D4, D4×D5, D4⋊2D5, C2×C5⋊D4, C20⋊2D4, D4⋊D10, D4.8D10, D20.23D4
►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)
(1 115)(2 114)(3 113)(4 112)(5 111)(6 110)(7 109)(8 108)(9 107)(10 106)(11 105)(12 104)(13 103)(14 102)(15 101)(16 120)(17 119)(18 118)(19 117)(20 116)(21 63)(22 62)(23 61)(24 80)(25 79)(26 78)(27 77)(28 76)(29 75)(30 74)(31 73)(32 72)(33 71)(34 70)(35 69)(36 68)(37 67)(38 66)(39 65)(40 64)(41 129)(42 128)(43 127)(44 126)(45 125)(46 124)(47 123)(48 122)(49 121)(50 140)(51 139)(52 138)(53 137)(54 136)(55 135)(56 134)(57 133)(58 132)(59 131)(60 130)(81 153)(82 152)(83 151)(84 150)(85 149)(86 148)(87 147)(88 146)(89 145)(90 144)(91 143)(92 142)(93 141)(94 160)(95 159)(96 158)(97 157)(98 156)(99 155)(100 154)
(1 69 116 36)(2 70 117 37)(3 71 118 38)(4 72 119 39)(5 73 120 40)(6 74 101 21)(7 75 102 22)(8 76 103 23)(9 77 104 24)(10 78 105 25)(11 79 106 26)(12 80 107 27)(13 61 108 28)(14 62 109 29)(15 63 110 30)(16 64 111 31)(17 65 112 32)(18 66 113 33)(19 67 114 34)(20 68 115 35)(41 88 127 144)(42 89 128 145)(43 90 129 146)(44 91 130 147)(45 92 131 148)(46 93 132 149)(47 94 133 150)(48 95 134 151)(49 96 135 152)(50 97 136 153)(51 98 137 154)(52 99 138 155)(53 100 139 156)(54 81 140 157)(55 82 121 158)(56 83 122 159)(57 84 123 160)(58 85 124 141)(59 86 125 142)(60 87 126 143)
(1 143 11 153)(2 154 12 144)(3 145 13 155)(4 156 14 146)(5 147 15 157)(6 158 16 148)(7 149 17 159)(8 160 18 150)(9 151 19 141)(10 142 20 152)(21 45 31 55)(22 56 32 46)(23 47 33 57)(24 58 34 48)(25 49 35 59)(26 60 36 50)(27 51 37 41)(28 42 38 52)(29 53 39 43)(30 44 40 54)(61 128 71 138)(62 139 72 129)(63 130 73 140)(64 121 74 131)(65 132 75 122)(66 123 76 133)(67 134 77 124)(68 125 78 135)(69 136 79 126)(70 127 80 137)(81 120 91 110)(82 111 92 101)(83 102 93 112)(84 113 94 103)(85 104 95 114)(86 115 96 105)(87 106 97 116)(88 117 98 107)(89 108 99 118)(90 119 100 109)
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), (1,115)(2,114)(3,113)(4,112)(5,111)(6,110)(7,109)(8,108)(9,107)(10,106)(11,105)(12,104)(13,103)(14,102)(15,101)(16,120)(17,119)(18,118)(19,117)(20,116)(21,63)(22,62)(23,61)(24,80)(25,79)(26,78)(27,77)(28,76)(29,75)(30,74)(31,73)(32,72)(33,71)(34,70)(35,69)(36,68)(37,67)(38,66)(39,65)(40,64)(41,129)(42,128)(43,127)(44,126)(45,125)(46,124)(47,123)(48,122)(49,121)(50,140)(51,139)(52,138)(53,137)(54,136)(55,135)(56,134)(57,133)(58,132)(59,131)(60,130)(81,153)(82,152)(83,151)(84,150)(85,149)(86,148)(87,147)(88,146)(89,145)(90,144)(91,143)(92,142)(93,141)(94,160)(95,159)(96,158)(97,157)(98,156)(99,155)(100,154), (1,69,116,36)(2,70,117,37)(3,71,118,38)(4,72,119,39)(5,73,120,40)(6,74,101,21)(7,75,102,22)(8,76,103,23)(9,77,104,24)(10,78,105,25)(11,79,106,26)(12,80,107,27)(13,61,108,28)(14,62,109,29)(15,63,110,30)(16,64,111,31)(17,65,112,32)(18,66,113,33)(19,67,114,34)(20,68,115,35)(41,88,127,144)(42,89,128,145)(43,90,129,146)(44,91,130,147)(45,92,131,148)(46,93,132,149)(47,94,133,150)(48,95,134,151)(49,96,135,152)(50,97,136,153)(51,98,137,154)(52,99,138,155)(53,100,139,156)(54,81,140,157)(55,82,121,158)(56,83,122,159)(57,84,123,160)(58,85,124,141)(59,86,125,142)(60,87,126,143), (1,143,11,153)(2,154,12,144)(3,145,13,155)(4,156,14,146)(5,147,15,157)(6,158,16,148)(7,149,17,159)(8,160,18,150)(9,151,19,141)(10,142,20,152)(21,45,31,55)(22,56,32,46)(23,47,33,57)(24,58,34,48)(25,49,35,59)(26,60,36,50)(27,51,37,41)(28,42,38,52)(29,53,39,43)(30,44,40,54)(61,128,71,138)(62,139,72,129)(63,130,73,140)(64,121,74,131)(65,132,75,122)(66,123,76,133)(67,134,77,124)(68,125,78,135)(69,136,79,126)(70,127,80,137)(81,120,91,110)(82,111,92,101)(83,102,93,112)(84,113,94,103)(85,104,95,114)(86,115,96,105)(87,106,97,116)(88,117,98,107)(89,108,99,118)(90,119,100,109)>;
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), (1,115)(2,114)(3,113)(4,112)(5,111)(6,110)(7,109)(8,108)(9,107)(10,106)(11,105)(12,104)(13,103)(14,102)(15,101)(16,120)(17,119)(18,118)(19,117)(20,116)(21,63)(22,62)(23,61)(24,80)(25,79)(26,78)(27,77)(28,76)(29,75)(30,74)(31,73)(32,72)(33,71)(34,70)(35,69)(36,68)(37,67)(38,66)(39,65)(40,64)(41,129)(42,128)(43,127)(44,126)(45,125)(46,124)(47,123)(48,122)(49,121)(50,140)(51,139)(52,138)(53,137)(54,136)(55,135)(56,134)(57,133)(58,132)(59,131)(60,130)(81,153)(82,152)(83,151)(84,150)(85,149)(86,148)(87,147)(88,146)(89,145)(90,144)(91,143)(92,142)(93,141)(94,160)(95,159)(96,158)(97,157)(98,156)(99,155)(100,154), (1,69,116,36)(2,70,117,37)(3,71,118,38)(4,72,119,39)(5,73,120,40)(6,74,101,21)(7,75,102,22)(8,76,103,23)(9,77,104,24)(10,78,105,25)(11,79,106,26)(12,80,107,27)(13,61,108,28)(14,62,109,29)(15,63,110,30)(16,64,111,31)(17,65,112,32)(18,66,113,33)(19,67,114,34)(20,68,115,35)(41,88,127,144)(42,89,128,145)(43,90,129,146)(44,91,130,147)(45,92,131,148)(46,93,132,149)(47,94,133,150)(48,95,134,151)(49,96,135,152)(50,97,136,153)(51,98,137,154)(52,99,138,155)(53,100,139,156)(54,81,140,157)(55,82,121,158)(56,83,122,159)(57,84,123,160)(58,85,124,141)(59,86,125,142)(60,87,126,143), (1,143,11,153)(2,154,12,144)(3,145,13,155)(4,156,14,146)(5,147,15,157)(6,158,16,148)(7,149,17,159)(8,160,18,150)(9,151,19,141)(10,142,20,152)(21,45,31,55)(22,56,32,46)(23,47,33,57)(24,58,34,48)(25,49,35,59)(26,60,36,50)(27,51,37,41)(28,42,38,52)(29,53,39,43)(30,44,40,54)(61,128,71,138)(62,139,72,129)(63,130,73,140)(64,121,74,131)(65,132,75,122)(66,123,76,133)(67,134,77,124)(68,125,78,135)(69,136,79,126)(70,127,80,137)(81,120,91,110)(82,111,92,101)(83,102,93,112)(84,113,94,103)(85,104,95,114)(86,115,96,105)(87,106,97,116)(88,117,98,107)(89,108,99,118)(90,119,100,109) );
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)], [(1,115),(2,114),(3,113),(4,112),(5,111),(6,110),(7,109),(8,108),(9,107),(10,106),(11,105),(12,104),(13,103),(14,102),(15,101),(16,120),(17,119),(18,118),(19,117),(20,116),(21,63),(22,62),(23,61),(24,80),(25,79),(26,78),(27,77),(28,76),(29,75),(30,74),(31,73),(32,72),(33,71),(34,70),(35,69),(36,68),(37,67),(38,66),(39,65),(40,64),(41,129),(42,128),(43,127),(44,126),(45,125),(46,124),(47,123),(48,122),(49,121),(50,140),(51,139),(52,138),(53,137),(54,136),(55,135),(56,134),(57,133),(58,132),(59,131),(60,130),(81,153),(82,152),(83,151),(84,150),(85,149),(86,148),(87,147),(88,146),(89,145),(90,144),(91,143),(92,142),(93,141),(94,160),(95,159),(96,158),(97,157),(98,156),(99,155),(100,154)], [(1,69,116,36),(2,70,117,37),(3,71,118,38),(4,72,119,39),(5,73,120,40),(6,74,101,21),(7,75,102,22),(8,76,103,23),(9,77,104,24),(10,78,105,25),(11,79,106,26),(12,80,107,27),(13,61,108,28),(14,62,109,29),(15,63,110,30),(16,64,111,31),(17,65,112,32),(18,66,113,33),(19,67,114,34),(20,68,115,35),(41,88,127,144),(42,89,128,145),(43,90,129,146),(44,91,130,147),(45,92,131,148),(46,93,132,149),(47,94,133,150),(48,95,134,151),(49,96,135,152),(50,97,136,153),(51,98,137,154),(52,99,138,155),(53,100,139,156),(54,81,140,157),(55,82,121,158),(56,83,122,159),(57,84,123,160),(58,85,124,141),(59,86,125,142),(60,87,126,143)], [(1,143,11,153),(2,154,12,144),(3,145,13,155),(4,156,14,146),(5,147,15,157),(6,158,16,148),(7,149,17,159),(8,160,18,150),(9,151,19,141),(10,142,20,152),(21,45,31,55),(22,56,32,46),(23,47,33,57),(24,58,34,48),(25,49,35,59),(26,60,36,50),(27,51,37,41),(28,42,38,52),(29,53,39,43),(30,44,40,54),(61,128,71,138),(62,139,72,129),(63,130,73,140),(64,121,74,131),(65,132,75,122),(66,123,76,133),(67,134,77,124),(68,125,78,135),(69,136,79,126),(70,127,80,137),(81,120,91,110),(82,111,92,101),(83,102,93,112),(84,113,94,103),(85,104,95,114),(86,115,96,105),(87,106,97,116),(88,117,98,107),(89,108,99,118),(90,119,100,109)])
47 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 | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 8 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 8 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | C5⋊D4 | C8⋊C22 | D4×D5 | D4⋊2D5 | D4⋊D10 | D4.8D10 |
| kernel | D20.23D4 | C20⋊3C8 | D4⋊Dic5 | Q8⋊Dic5 | C4×D20 | C2×D4⋊D5 | C2×Q8⋊D5 | C5×C4.4D4 | D20 | C2×C20 | C4.4D4 | C20 | C42 | C2×D4 | C2×Q8 | C10 | C2×C4 | C10 | C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D20.23D4 ►in GL6(𝔽41)
| 35 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 38 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 18 | 6 | 0 | 0 | 0 | 0 |
| 35 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 15 | 0 | 0 |
| 0 | 0 | 15 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 13 |
| 0 | 0 | 0 | 0 | 16 | 11 |
G:=sub<GL(6,GF(41))| [35,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,38,32],[18,35,0,0,0,0,6,23,0,0,0,0,0,0,15,15,0,0,0,0,15,26,0,0,0,0,0,0,30,16,0,0,0,0,13,11] >;
D20.23D4 in GAP, Magma, Sage, TeX
D_{20}._{23}D_4 % in TeX
G:=Group("D20.23D4"); // GroupNames label
G:=SmallGroup(320,684);
// by ID
G=gap.SmallGroup(320,684);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,344,254,219,1123,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=a^10,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^15*b,d*c*d^-1=a^10*c^-1>;
// generators/relations