metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊14D4, Dic10⋊13D4, C23.12D20, (C2×D40)⋊3C2, C22⋊C8⋊6D5, (C2×C8).3D10, C5⋊1(D4⋊D4), C4.122(D4×D5), D20⋊5C4⋊9C2, C20⋊7D4⋊16C2, C10.9(C4○D8), (C2×C20).241D4, (C2×C4).119D20, C20.334(C2×D4), C10.10C22≀C2, C20.44D4⋊6C2, (C22×C10).54D4, (C22×C4).84D10, C2.13(C8⋊D10), C10.10(C8⋊C22), (C2×C20).744C23, (C2×C40).119C22, C22.107(C2×D20), C4⋊Dic5.13C22, C2.13(C22⋊D20), C2.11(D40⋊7C2), (C2×D20).199C22, (C22×C20).97C22, (C2×Dic10).217C22, (C2×C4○D20)⋊1C2, (C5×C22⋊C8)⋊8C2, (C2×C40⋊C2)⋊11C2, (C2×C10).127(C2×D4), (C2×C4).689(C22×D5), SmallGroup(320,361)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D20⋊14D4
G = < a,b,c,d | a20=b2=c4=d2=1, bab=cac-1=a-1, ad=da, cbc-1=a13b, dbd=a10b, dcd=c-1 >
Subgroups: 830 in 162 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, D4⋊D4, C40⋊C2, D40, C4⋊Dic5, D10⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, C20.44D4, D20⋊5C4, C5×C22⋊C8, C2×C40⋊C2, C2×D40, C20⋊7D4, C2×C4○D20, D20⋊14D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C4○D8, C8⋊C22, D20, C22×D5, D4⋊D4, C2×D20, D4×D5, C22⋊D20, D40⋊7C2, C8⋊D10, D20⋊14D4
►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)
(1 153)(2 152)(3 151)(4 150)(5 149)(6 148)(7 147)(8 146)(9 145)(10 144)(11 143)(12 142)(13 141)(14 160)(15 159)(16 158)(17 157)(18 156)(19 155)(20 154)(21 64)(22 63)(23 62)(24 61)(25 80)(26 79)(27 78)(28 77)(29 76)(30 75)(31 74)(32 73)(33 72)(34 71)(35 70)(36 69)(37 68)(38 67)(39 66)(40 65)(41 106)(42 105)(43 104)(44 103)(45 102)(46 101)(47 120)(48 119)(49 118)(50 117)(51 116)(52 115)(53 114)(54 113)(55 112)(56 111)(57 110)(58 109)(59 108)(60 107)(81 128)(82 127)(83 126)(84 125)(85 124)(86 123)(87 122)(88 121)(89 140)(90 139)(91 138)(92 137)(93 136)(94 135)(95 134)(96 133)(97 132)(98 131)(99 130)(100 129)
(1 71 92 50)(2 70 93 49)(3 69 94 48)(4 68 95 47)(5 67 96 46)(6 66 97 45)(7 65 98 44)(8 64 99 43)(9 63 100 42)(10 62 81 41)(11 61 82 60)(12 80 83 59)(13 79 84 58)(14 78 85 57)(15 77 86 56)(16 76 87 55)(17 75 88 54)(18 74 89 53)(19 73 90 52)(20 72 91 51)(21 137 104 153)(22 136 105 152)(23 135 106 151)(24 134 107 150)(25 133 108 149)(26 132 109 148)(27 131 110 147)(28 130 111 146)(29 129 112 145)(30 128 113 144)(31 127 114 143)(32 126 115 142)(33 125 116 141)(34 124 117 160)(35 123 118 159)(36 122 119 158)(37 121 120 157)(38 140 101 156)(39 139 102 155)(40 138 103 154)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 134)(22 135)(23 136)(24 137)(25 138)(26 139)(27 140)(28 121)(29 122)(30 123)(31 124)(32 125)(33 126)(34 127)(35 128)(36 129)(37 130)(38 131)(39 132)(40 133)(61 82)(62 83)(63 84)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 99)(79 100)(80 81)(101 147)(102 148)(103 149)(104 150)(105 151)(106 152)(107 153)(108 154)(109 155)(110 156)(111 157)(112 158)(113 159)(114 160)(115 141)(116 142)(117 143)(118 144)(119 145)(120 146)
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), (1,153)(2,152)(3,151)(4,150)(5,149)(6,148)(7,147)(8,146)(9,145)(10,144)(11,143)(12,142)(13,141)(14,160)(15,159)(16,158)(17,157)(18,156)(19,155)(20,154)(21,64)(22,63)(23,62)(24,61)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,106)(42,105)(43,104)(44,103)(45,102)(46,101)(47,120)(48,119)(49,118)(50,117)(51,116)(52,115)(53,114)(54,113)(55,112)(56,111)(57,110)(58,109)(59,108)(60,107)(81,128)(82,127)(83,126)(84,125)(85,124)(86,123)(87,122)(88,121)(89,140)(90,139)(91,138)(92,137)(93,136)(94,135)(95,134)(96,133)(97,132)(98,131)(99,130)(100,129), (1,71,92,50)(2,70,93,49)(3,69,94,48)(4,68,95,47)(5,67,96,46)(6,66,97,45)(7,65,98,44)(8,64,99,43)(9,63,100,42)(10,62,81,41)(11,61,82,60)(12,80,83,59)(13,79,84,58)(14,78,85,57)(15,77,86,56)(16,76,87,55)(17,75,88,54)(18,74,89,53)(19,73,90,52)(20,72,91,51)(21,137,104,153)(22,136,105,152)(23,135,106,151)(24,134,107,150)(25,133,108,149)(26,132,109,148)(27,131,110,147)(28,130,111,146)(29,129,112,145)(30,128,113,144)(31,127,114,143)(32,126,115,142)(33,125,116,141)(34,124,117,160)(35,123,118,159)(36,122,119,158)(37,121,120,157)(38,140,101,156)(39,139,102,155)(40,138,103,154), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,134)(22,135)(23,136)(24,137)(25,138)(26,139)(27,140)(28,121)(29,122)(30,123)(31,124)(32,125)(33,126)(34,127)(35,128)(36,129)(37,130)(38,131)(39,132)(40,133)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,81)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,159)(114,160)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146)>;
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), (1,153)(2,152)(3,151)(4,150)(5,149)(6,148)(7,147)(8,146)(9,145)(10,144)(11,143)(12,142)(13,141)(14,160)(15,159)(16,158)(17,157)(18,156)(19,155)(20,154)(21,64)(22,63)(23,62)(24,61)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,106)(42,105)(43,104)(44,103)(45,102)(46,101)(47,120)(48,119)(49,118)(50,117)(51,116)(52,115)(53,114)(54,113)(55,112)(56,111)(57,110)(58,109)(59,108)(60,107)(81,128)(82,127)(83,126)(84,125)(85,124)(86,123)(87,122)(88,121)(89,140)(90,139)(91,138)(92,137)(93,136)(94,135)(95,134)(96,133)(97,132)(98,131)(99,130)(100,129), (1,71,92,50)(2,70,93,49)(3,69,94,48)(4,68,95,47)(5,67,96,46)(6,66,97,45)(7,65,98,44)(8,64,99,43)(9,63,100,42)(10,62,81,41)(11,61,82,60)(12,80,83,59)(13,79,84,58)(14,78,85,57)(15,77,86,56)(16,76,87,55)(17,75,88,54)(18,74,89,53)(19,73,90,52)(20,72,91,51)(21,137,104,153)(22,136,105,152)(23,135,106,151)(24,134,107,150)(25,133,108,149)(26,132,109,148)(27,131,110,147)(28,130,111,146)(29,129,112,145)(30,128,113,144)(31,127,114,143)(32,126,115,142)(33,125,116,141)(34,124,117,160)(35,123,118,159)(36,122,119,158)(37,121,120,157)(38,140,101,156)(39,139,102,155)(40,138,103,154), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,57)(9,58)(10,59)(11,60)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,49)(21,134)(22,135)(23,136)(24,137)(25,138)(26,139)(27,140)(28,121)(29,122)(30,123)(31,124)(32,125)(33,126)(34,127)(35,128)(36,129)(37,130)(38,131)(39,132)(40,133)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,81)(101,147)(102,148)(103,149)(104,150)(105,151)(106,152)(107,153)(108,154)(109,155)(110,156)(111,157)(112,158)(113,159)(114,160)(115,141)(116,142)(117,143)(118,144)(119,145)(120,146) );
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)], [(1,153),(2,152),(3,151),(4,150),(5,149),(6,148),(7,147),(8,146),(9,145),(10,144),(11,143),(12,142),(13,141),(14,160),(15,159),(16,158),(17,157),(18,156),(19,155),(20,154),(21,64),(22,63),(23,62),(24,61),(25,80),(26,79),(27,78),(28,77),(29,76),(30,75),(31,74),(32,73),(33,72),(34,71),(35,70),(36,69),(37,68),(38,67),(39,66),(40,65),(41,106),(42,105),(43,104),(44,103),(45,102),(46,101),(47,120),(48,119),(49,118),(50,117),(51,116),(52,115),(53,114),(54,113),(55,112),(56,111),(57,110),(58,109),(59,108),(60,107),(81,128),(82,127),(83,126),(84,125),(85,124),(86,123),(87,122),(88,121),(89,140),(90,139),(91,138),(92,137),(93,136),(94,135),(95,134),(96,133),(97,132),(98,131),(99,130),(100,129)], [(1,71,92,50),(2,70,93,49),(3,69,94,48),(4,68,95,47),(5,67,96,46),(6,66,97,45),(7,65,98,44),(8,64,99,43),(9,63,100,42),(10,62,81,41),(11,61,82,60),(12,80,83,59),(13,79,84,58),(14,78,85,57),(15,77,86,56),(16,76,87,55),(17,75,88,54),(18,74,89,53),(19,73,90,52),(20,72,91,51),(21,137,104,153),(22,136,105,152),(23,135,106,151),(24,134,107,150),(25,133,108,149),(26,132,109,148),(27,131,110,147),(28,130,111,146),(29,129,112,145),(30,128,113,144),(31,127,114,143),(32,126,115,142),(33,125,116,141),(34,124,117,160),(35,123,118,159),(36,122,119,158),(37,121,120,157),(38,140,101,156),(39,139,102,155),(40,138,103,154)], [(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,57),(9,58),(10,59),(11,60),(12,41),(13,42),(14,43),(15,44),(16,45),(17,46),(18,47),(19,48),(20,49),(21,134),(22,135),(23,136),(24,137),(25,138),(26,139),(27,140),(28,121),(29,122),(30,123),(31,124),(32,125),(33,126),(34,127),(35,128),(36,129),(37,130),(38,131),(39,132),(40,133),(61,82),(62,83),(63,84),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(71,92),(72,93),(73,94),(74,95),(75,96),(76,97),(77,98),(78,99),(79,100),(80,81),(101,147),(102,148),(103,149),(104,150),(105,151),(106,152),(107,153),(108,154),(109,155),(110,156),(111,157),(112,158),(113,159),(114,160),(115,141),(116,142),(117,143),(118,144),(119,145),(120,146)]])
59 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 20 | 20 | 40 | 2 | 2 | 2 | 2 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
59 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D4 | D5 | D10 | D10 | C4○D8 | D20 | D20 | D40⋊7C2 | C8⋊C22 | D4×D5 | C8⋊D10 |
| kernel | D20⋊14D4 | C20.44D4 | D20⋊5C4 | C5×C22⋊C8 | C2×C40⋊C2 | C2×D40 | C20⋊7D4 | C2×C4○D20 | Dic10 | D20 | C2×C20 | C22×C10 | C22⋊C8 | C2×C8 | C22×C4 | C10 | C2×C4 | C23 | C2 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 16 | 1 | 4 | 4 |
Matrix representation of D20⋊14D4 ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 29 | 29 | 0 | 0 | 0 | 0 |
| 29 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 39 | 0 | 0 |
| 0 | 0 | 2 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 4 |
| 0 | 0 | 0 | 0 | 12 | 19 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 4 |
| 0 | 0 | 0 | 0 | 33 | 19 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[29,29,0,0,0,0,29,12,0,0,0,0,0,0,28,2,0,0,0,0,39,13,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,40,35,0,0,0,0,0,1,0,0,0,0,0,0,22,12,0,0,0,0,4,19],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,22,33,0,0,0,0,4,19] >;
D20⋊14D4 in GAP, Magma, Sage, TeX
D_{20}\rtimes_{14}D_4 % in TeX
G:=Group("D20:14D4"); // GroupNames label
G:=SmallGroup(320,361);
// by ID
G=gap.SmallGroup(320,361);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,226,1123,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^20=b^2=c^4=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^13*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations