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, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C5⋊2C8, C4×D5, D20, D20, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, D4.2D4, C2×C5⋊2C8, C4⋊Dic5, D10⋊C4, D4⋊D5, Q8⋊D5, C4×C20, C5×C22⋊C4, 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, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C4○D8, C8⋊C22, C5⋊D4, 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)
(1 121)(2 140)(3 139)(4 138)(5 137)(6 136)(7 135)(8 134)(9 133)(10 132)(11 131)(12 130)(13 129)(14 128)(15 127)(16 126)(17 125)(18 124)(19 123)(20 122)(21 148)(22 147)(23 146)(24 145)(25 144)(26 143)(27 142)(28 141)(29 160)(30 159)(31 158)(32 157)(33 156)(34 155)(35 154)(36 153)(37 152)(38 151)(39 150)(40 149)(41 101)(42 120)(43 119)(44 118)(45 117)(46 116)(47 115)(48 114)(49 113)(50 112)(51 111)(52 110)(53 109)(54 108)(55 107)(56 106)(57 105)(58 104)(59 103)(60 102)(61 89)(62 88)(63 87)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(70 100)(71 99)(72 98)(73 97)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)
(1 59 122 104)(2 60 123 105)(3 41 124 106)(4 42 125 107)(5 43 126 108)(6 44 127 109)(7 45 128 110)(8 46 129 111)(9 47 130 112)(10 48 131 113)(11 49 132 114)(12 50 133 115)(13 51 134 116)(14 52 135 117)(15 53 136 118)(16 54 137 119)(17 55 138 120)(18 56 139 101)(19 57 140 102)(20 58 121 103)(21 94 154 62)(22 95 155 63)(23 96 156 64)(24 97 157 65)(25 98 158 66)(26 99 159 67)(27 100 160 68)(28 81 141 69)(29 82 142 70)(30 83 143 71)(31 84 144 72)(32 85 145 73)(33 86 146 74)(34 87 147 75)(35 88 148 76)(36 89 149 77)(37 90 150 78)(38 91 151 79)(39 92 152 80)(40 93 153 61)
(1 89 11 99)(2 100 12 90)(3 91 13 81)(4 82 14 92)(5 93 15 83)(6 84 16 94)(7 95 17 85)(8 86 18 96)(9 97 19 87)(10 88 20 98)(21 54 31 44)(22 45 32 55)(23 56 33 46)(24 47 34 57)(25 58 35 48)(26 49 36 59)(27 60 37 50)(28 51 38 41)(29 42 39 52)(30 53 40 43)(61 136 71 126)(62 127 72 137)(63 138 73 128)(64 129 74 139)(65 140 75 130)(66 131 76 121)(67 122 77 132)(68 133 78 123)(69 124 79 134)(70 135 80 125)(101 146 111 156)(102 157 112 147)(103 148 113 158)(104 159 114 149)(105 150 115 160)(106 141 116 151)(107 152 117 142)(108 143 118 153)(109 154 119 144)(110 145 120 155)
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), (1,121)(2,140)(3,139)(4,138)(5,137)(6,136)(7,135)(8,134)(9,133)(10,132)(11,131)(12,130)(13,129)(14,128)(15,127)(16,126)(17,125)(18,124)(19,123)(20,122)(21,148)(22,147)(23,146)(24,145)(25,144)(26,143)(27,142)(28,141)(29,160)(30,159)(31,158)(32,157)(33,156)(34,155)(35,154)(36,153)(37,152)(38,151)(39,150)(40,149)(41,101)(42,120)(43,119)(44,118)(45,117)(46,116)(47,115)(48,114)(49,113)(50,112)(51,111)(52,110)(53,109)(54,108)(55,107)(56,106)(57,105)(58,104)(59,103)(60,102)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90), (1,59,122,104)(2,60,123,105)(3,41,124,106)(4,42,125,107)(5,43,126,108)(6,44,127,109)(7,45,128,110)(8,46,129,111)(9,47,130,112)(10,48,131,113)(11,49,132,114)(12,50,133,115)(13,51,134,116)(14,52,135,117)(15,53,136,118)(16,54,137,119)(17,55,138,120)(18,56,139,101)(19,57,140,102)(20,58,121,103)(21,94,154,62)(22,95,155,63)(23,96,156,64)(24,97,157,65)(25,98,158,66)(26,99,159,67)(27,100,160,68)(28,81,141,69)(29,82,142,70)(30,83,143,71)(31,84,144,72)(32,85,145,73)(33,86,146,74)(34,87,147,75)(35,88,148,76)(36,89,149,77)(37,90,150,78)(38,91,151,79)(39,92,152,80)(40,93,153,61), (1,89,11,99)(2,100,12,90)(3,91,13,81)(4,82,14,92)(5,93,15,83)(6,84,16,94)(7,95,17,85)(8,86,18,96)(9,97,19,87)(10,88,20,98)(21,54,31,44)(22,45,32,55)(23,56,33,46)(24,47,34,57)(25,58,35,48)(26,49,36,59)(27,60,37,50)(28,51,38,41)(29,42,39,52)(30,53,40,43)(61,136,71,126)(62,127,72,137)(63,138,73,128)(64,129,74,139)(65,140,75,130)(66,131,76,121)(67,122,77,132)(68,133,78,123)(69,124,79,134)(70,135,80,125)(101,146,111,156)(102,157,112,147)(103,148,113,158)(104,159,114,149)(105,150,115,160)(106,141,116,151)(107,152,117,142)(108,143,118,153)(109,154,119,144)(110,145,120,155)>;
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), (1,121)(2,140)(3,139)(4,138)(5,137)(6,136)(7,135)(8,134)(9,133)(10,132)(11,131)(12,130)(13,129)(14,128)(15,127)(16,126)(17,125)(18,124)(19,123)(20,122)(21,148)(22,147)(23,146)(24,145)(25,144)(26,143)(27,142)(28,141)(29,160)(30,159)(31,158)(32,157)(33,156)(34,155)(35,154)(36,153)(37,152)(38,151)(39,150)(40,149)(41,101)(42,120)(43,119)(44,118)(45,117)(46,116)(47,115)(48,114)(49,113)(50,112)(51,111)(52,110)(53,109)(54,108)(55,107)(56,106)(57,105)(58,104)(59,103)(60,102)(61,89)(62,88)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,100)(71,99)(72,98)(73,97)(74,96)(75,95)(76,94)(77,93)(78,92)(79,91)(80,90), (1,59,122,104)(2,60,123,105)(3,41,124,106)(4,42,125,107)(5,43,126,108)(6,44,127,109)(7,45,128,110)(8,46,129,111)(9,47,130,112)(10,48,131,113)(11,49,132,114)(12,50,133,115)(13,51,134,116)(14,52,135,117)(15,53,136,118)(16,54,137,119)(17,55,138,120)(18,56,139,101)(19,57,140,102)(20,58,121,103)(21,94,154,62)(22,95,155,63)(23,96,156,64)(24,97,157,65)(25,98,158,66)(26,99,159,67)(27,100,160,68)(28,81,141,69)(29,82,142,70)(30,83,143,71)(31,84,144,72)(32,85,145,73)(33,86,146,74)(34,87,147,75)(35,88,148,76)(36,89,149,77)(37,90,150,78)(38,91,151,79)(39,92,152,80)(40,93,153,61), (1,89,11,99)(2,100,12,90)(3,91,13,81)(4,82,14,92)(5,93,15,83)(6,84,16,94)(7,95,17,85)(8,86,18,96)(9,97,19,87)(10,88,20,98)(21,54,31,44)(22,45,32,55)(23,56,33,46)(24,47,34,57)(25,58,35,48)(26,49,36,59)(27,60,37,50)(28,51,38,41)(29,42,39,52)(30,53,40,43)(61,136,71,126)(62,127,72,137)(63,138,73,128)(64,129,74,139)(65,140,75,130)(66,131,76,121)(67,122,77,132)(68,133,78,123)(69,124,79,134)(70,135,80,125)(101,146,111,156)(102,157,112,147)(103,148,113,158)(104,159,114,149)(105,150,115,160)(106,141,116,151)(107,152,117,142)(108,143,118,153)(109,154,119,144)(110,145,120,155) );
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)], [(1,121),(2,140),(3,139),(4,138),(5,137),(6,136),(7,135),(8,134),(9,133),(10,132),(11,131),(12,130),(13,129),(14,128),(15,127),(16,126),(17,125),(18,124),(19,123),(20,122),(21,148),(22,147),(23,146),(24,145),(25,144),(26,143),(27,142),(28,141),(29,160),(30,159),(31,158),(32,157),(33,156),(34,155),(35,154),(36,153),(37,152),(38,151),(39,150),(40,149),(41,101),(42,120),(43,119),(44,118),(45,117),(46,116),(47,115),(48,114),(49,113),(50,112),(51,111),(52,110),(53,109),(54,108),(55,107),(56,106),(57,105),(58,104),(59,103),(60,102),(61,89),(62,88),(63,87),(64,86),(65,85),(66,84),(67,83),(68,82),(69,81),(70,100),(71,99),(72,98),(73,97),(74,96),(75,95),(76,94),(77,93),(78,92),(79,91),(80,90)], [(1,59,122,104),(2,60,123,105),(3,41,124,106),(4,42,125,107),(5,43,126,108),(6,44,127,109),(7,45,128,110),(8,46,129,111),(9,47,130,112),(10,48,131,113),(11,49,132,114),(12,50,133,115),(13,51,134,116),(14,52,135,117),(15,53,136,118),(16,54,137,119),(17,55,138,120),(18,56,139,101),(19,57,140,102),(20,58,121,103),(21,94,154,62),(22,95,155,63),(23,96,156,64),(24,97,157,65),(25,98,158,66),(26,99,159,67),(27,100,160,68),(28,81,141,69),(29,82,142,70),(30,83,143,71),(31,84,144,72),(32,85,145,73),(33,86,146,74),(34,87,147,75),(35,88,148,76),(36,89,149,77),(37,90,150,78),(38,91,151,79),(39,92,152,80),(40,93,153,61)], [(1,89,11,99),(2,100,12,90),(3,91,13,81),(4,82,14,92),(5,93,15,83),(6,84,16,94),(7,95,17,85),(8,86,18,96),(9,97,19,87),(10,88,20,98),(21,54,31,44),(22,45,32,55),(23,56,33,46),(24,47,34,57),(25,58,35,48),(26,49,36,59),(27,60,37,50),(28,51,38,41),(29,42,39,52),(30,53,40,43),(61,136,71,126),(62,127,72,137),(63,138,73,128),(64,129,74,139),(65,140,75,130),(66,131,76,121),(67,122,77,132),(68,133,78,123),(69,124,79,134),(70,135,80,125),(101,146,111,156),(102,157,112,147),(103,148,113,158),(104,159,114,149),(105,150,115,160),(106,141,116,151),(107,152,117,142),(108,143,118,153),(109,154,119,144),(110,145,120,155)]])
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