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 [×3], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×10], C5, C8 [×2], C2×C4 [×2], C2×C4 [×8], D4 [×11], Q8 [×3], C23, C23 [×2], D5 [×3], C10 [×3], C10, C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×2], SD16 [×2], C22×C4, C22×C4, C2×D4 [×4], C2×Q8, C4○D4 [×4], Dic5 [×3], C20 [×2], C20, D10 [×7], C2×C10, C2×C10 [×3], C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C40 [×2], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20 [×3], C2×Dic5 [×2], C5⋊D4 [×6], C2×C20 [×2], C2×C20 [×2], C22×D5 [×2], C22×C10, D4⋊D4, C40⋊C2 [×2], D40 [×2], C4⋊Dic5, D10⋊C4, C2×C40 [×2], C2×Dic10, C2×C4×D5, C2×D20 [×2], C4○D20 [×4], C2×C5⋊D4 [×2], C22×C20, C20.44D4, D20⋊5C4, C5×C22⋊C8, C2×C40⋊C2, C2×D40, C20⋊7D4, C2×C4○D20, D20⋊14D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C4○D8, C8⋊C22, D20 [×2], C22×D5, D4⋊D4, C2×D20, D4×D5 [×2], 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 92)(2 91)(3 90)(4 89)(5 88)(6 87)(7 86)(8 85)(9 84)(10 83)(11 82)(12 81)(13 100)(14 99)(15 98)(16 97)(17 96)(18 95)(19 94)(20 93)(21 121)(22 140)(23 139)(24 138)(25 137)(26 136)(27 135)(28 134)(29 133)(30 132)(31 131)(32 130)(33 129)(34 128)(35 127)(36 126)(37 125)(38 124)(39 123)(40 122)(41 108)(42 107)(43 106)(44 105)(45 104)(46 103)(47 102)(48 101)(49 120)(50 119)(51 118)(52 117)(53 116)(54 115)(55 114)(56 113)(57 112)(58 111)(59 110)(60 109)(61 160)(62 159)(63 158)(64 157)(65 156)(66 155)(67 154)(68 153)(69 152)(70 151)(71 150)(72 149)(73 148)(74 147)(75 146)(76 145)(77 144)(78 143)(79 142)(80 141)
(1 77 103 30)(2 76 104 29)(3 75 105 28)(4 74 106 27)(5 73 107 26)(6 72 108 25)(7 71 109 24)(8 70 110 23)(9 69 111 22)(10 68 112 21)(11 67 113 40)(12 66 114 39)(13 65 115 38)(14 64 116 37)(15 63 117 36)(16 62 118 35)(17 61 119 34)(18 80 120 33)(19 79 101 32)(20 78 102 31)(41 124 87 156)(42 123 88 155)(43 122 89 154)(44 121 90 153)(45 140 91 152)(46 139 92 151)(47 138 93 150)(48 137 94 149)(49 136 95 148)(50 135 96 147)(51 134 97 146)(52 133 98 145)(53 132 99 144)(54 131 100 143)(55 130 81 142)(56 129 82 141)(57 128 83 160)(58 127 84 159)(59 126 85 158)(60 125 86 157)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 21)(13 22)(14 23)(15 24)(16 25)(17 26)(18 27)(19 28)(20 29)(41 149)(42 150)(43 151)(44 152)(45 153)(46 154)(47 155)(48 156)(49 157)(50 158)(51 159)(52 160)(53 141)(54 142)(55 143)(56 144)(57 145)(58 146)(59 147)(60 148)(61 107)(62 108)(63 109)(64 110)(65 111)(66 112)(67 113)(68 114)(69 115)(70 116)(71 117)(72 118)(73 119)(74 120)(75 101)(76 102)(77 103)(78 104)(79 105)(80 106)(81 131)(82 132)(83 133)(84 134)(85 135)(86 136)(87 137)(88 138)(89 139)(90 140)(91 121)(92 122)(93 123)(94 124)(95 125)(96 126)(97 127)(98 128)(99 129)(100 130)
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), (1,92)(2,91)(3,90)(4,89)(5,88)(6,87)(7,86)(8,85)(9,84)(10,83)(11,82)(12,81)(13,100)(14,99)(15,98)(16,97)(17,96)(18,95)(19,94)(20,93)(21,121)(22,140)(23,139)(24,138)(25,137)(26,136)(27,135)(28,134)(29,133)(30,132)(31,131)(32,130)(33,129)(34,128)(35,127)(36,126)(37,125)(38,124)(39,123)(40,122)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,112)(58,111)(59,110)(60,109)(61,160)(62,159)(63,158)(64,157)(65,156)(66,155)(67,154)(68,153)(69,152)(70,151)(71,150)(72,149)(73,148)(74,147)(75,146)(76,145)(77,144)(78,143)(79,142)(80,141), (1,77,103,30)(2,76,104,29)(3,75,105,28)(4,74,106,27)(5,73,107,26)(6,72,108,25)(7,71,109,24)(8,70,110,23)(9,69,111,22)(10,68,112,21)(11,67,113,40)(12,66,114,39)(13,65,115,38)(14,64,116,37)(15,63,117,36)(16,62,118,35)(17,61,119,34)(18,80,120,33)(19,79,101,32)(20,78,102,31)(41,124,87,156)(42,123,88,155)(43,122,89,154)(44,121,90,153)(45,140,91,152)(46,139,92,151)(47,138,93,150)(48,137,94,149)(49,136,95,148)(50,135,96,147)(51,134,97,146)(52,133,98,145)(53,132,99,144)(54,131,100,143)(55,130,81,142)(56,129,82,141)(57,128,83,160)(58,127,84,159)(59,126,85,158)(60,125,86,157), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(41,149)(42,150)(43,151)(44,152)(45,153)(46,154)(47,155)(48,156)(49,157)(50,158)(51,159)(52,160)(53,141)(54,142)(55,143)(56,144)(57,145)(58,146)(59,147)(60,148)(61,107)(62,108)(63,109)(64,110)(65,111)(66,112)(67,113)(68,114)(69,115)(70,116)(71,117)(72,118)(73,119)(74,120)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,131)(82,132)(83,133)(84,134)(85,135)(86,136)(87,137)(88,138)(89,139)(90,140)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(97,127)(98,128)(99,129)(100,130)>;
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), (1,92)(2,91)(3,90)(4,89)(5,88)(6,87)(7,86)(8,85)(9,84)(10,83)(11,82)(12,81)(13,100)(14,99)(15,98)(16,97)(17,96)(18,95)(19,94)(20,93)(21,121)(22,140)(23,139)(24,138)(25,137)(26,136)(27,135)(28,134)(29,133)(30,132)(31,131)(32,130)(33,129)(34,128)(35,127)(36,126)(37,125)(38,124)(39,123)(40,122)(41,108)(42,107)(43,106)(44,105)(45,104)(46,103)(47,102)(48,101)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,112)(58,111)(59,110)(60,109)(61,160)(62,159)(63,158)(64,157)(65,156)(66,155)(67,154)(68,153)(69,152)(70,151)(71,150)(72,149)(73,148)(74,147)(75,146)(76,145)(77,144)(78,143)(79,142)(80,141), (1,77,103,30)(2,76,104,29)(3,75,105,28)(4,74,106,27)(5,73,107,26)(6,72,108,25)(7,71,109,24)(8,70,110,23)(9,69,111,22)(10,68,112,21)(11,67,113,40)(12,66,114,39)(13,65,115,38)(14,64,116,37)(15,63,117,36)(16,62,118,35)(17,61,119,34)(18,80,120,33)(19,79,101,32)(20,78,102,31)(41,124,87,156)(42,123,88,155)(43,122,89,154)(44,121,90,153)(45,140,91,152)(46,139,92,151)(47,138,93,150)(48,137,94,149)(49,136,95,148)(50,135,96,147)(51,134,97,146)(52,133,98,145)(53,132,99,144)(54,131,100,143)(55,130,81,142)(56,129,82,141)(57,128,83,160)(58,127,84,159)(59,126,85,158)(60,125,86,157), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(19,28)(20,29)(41,149)(42,150)(43,151)(44,152)(45,153)(46,154)(47,155)(48,156)(49,157)(50,158)(51,159)(52,160)(53,141)(54,142)(55,143)(56,144)(57,145)(58,146)(59,147)(60,148)(61,107)(62,108)(63,109)(64,110)(65,111)(66,112)(67,113)(68,114)(69,115)(70,116)(71,117)(72,118)(73,119)(74,120)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,131)(82,132)(83,133)(84,134)(85,135)(86,136)(87,137)(88,138)(89,139)(90,140)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(97,127)(98,128)(99,129)(100,130) );
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)], [(1,92),(2,91),(3,90),(4,89),(5,88),(6,87),(7,86),(8,85),(9,84),(10,83),(11,82),(12,81),(13,100),(14,99),(15,98),(16,97),(17,96),(18,95),(19,94),(20,93),(21,121),(22,140),(23,139),(24,138),(25,137),(26,136),(27,135),(28,134),(29,133),(30,132),(31,131),(32,130),(33,129),(34,128),(35,127),(36,126),(37,125),(38,124),(39,123),(40,122),(41,108),(42,107),(43,106),(44,105),(45,104),(46,103),(47,102),(48,101),(49,120),(50,119),(51,118),(52,117),(53,116),(54,115),(55,114),(56,113),(57,112),(58,111),(59,110),(60,109),(61,160),(62,159),(63,158),(64,157),(65,156),(66,155),(67,154),(68,153),(69,152),(70,151),(71,150),(72,149),(73,148),(74,147),(75,146),(76,145),(77,144),(78,143),(79,142),(80,141)], [(1,77,103,30),(2,76,104,29),(3,75,105,28),(4,74,106,27),(5,73,107,26),(6,72,108,25),(7,71,109,24),(8,70,110,23),(9,69,111,22),(10,68,112,21),(11,67,113,40),(12,66,114,39),(13,65,115,38),(14,64,116,37),(15,63,117,36),(16,62,118,35),(17,61,119,34),(18,80,120,33),(19,79,101,32),(20,78,102,31),(41,124,87,156),(42,123,88,155),(43,122,89,154),(44,121,90,153),(45,140,91,152),(46,139,92,151),(47,138,93,150),(48,137,94,149),(49,136,95,148),(50,135,96,147),(51,134,97,146),(52,133,98,145),(53,132,99,144),(54,131,100,143),(55,130,81,142),(56,129,82,141),(57,128,83,160),(58,127,84,159),(59,126,85,158),(60,125,86,157)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,40),(12,21),(13,22),(14,23),(15,24),(16,25),(17,26),(18,27),(19,28),(20,29),(41,149),(42,150),(43,151),(44,152),(45,153),(46,154),(47,155),(48,156),(49,157),(50,158),(51,159),(52,160),(53,141),(54,142),(55,143),(56,144),(57,145),(58,146),(59,147),(60,148),(61,107),(62,108),(63,109),(64,110),(65,111),(66,112),(67,113),(68,114),(69,115),(70,116),(71,117),(72,118),(73,119),(74,120),(75,101),(76,102),(77,103),(78,104),(79,105),(80,106),(81,131),(82,132),(83,133),(84,134),(85,135),(86,136),(87,137),(88,138),(89,139),(90,140),(91,121),(92,122),(93,123),(94,124),(95,125),(96,126),(97,127),(98,128),(99,129),(100,130)])
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