metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D16⋊3D5, D8.1D10, D10.5D8, C16.8D10, Dic40⋊4C2, C80.6C22, C40.15C23, Dic5.24D8, Dic20.2C22, C4.3(D4×D5), (D5×C16)⋊2C2, (C5×D16)⋊3C2, C5⋊2(C4○D16), D8.D5⋊2C2, C2.18(D5×D8), C20.9(C2×D4), D8⋊3D5⋊4C2, (C4×D5).58D4, C10.34(C2×D8), C5⋊2C8.24D4, (C5×D8).1C22, C8.21(C22×D5), C5⋊2C16.5C22, (C8×D5).39C22, SmallGroup(320,539)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D16⋊3D5
G = < a,b,c,d | a16=b2=c5=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a8b, dcd=c-1 >
Subgroups: 390 in 84 conjugacy classes, 31 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, Q8, D5, C10, C10, C16, C16, C2×C8, D8, SD16, Q16, C4○D4, Dic5, Dic5, C20, D10, C2×C10, C2×C16, D16, SD32, Q32, C4○D8, C5⋊2C8, C40, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C4○D16, C5⋊2C16, C80, C8×D5, Dic20, D4.D5, C5×D8, D4⋊2D5, D5×C16, Dic40, D8.D5, C5×D16, D8⋊3D5, D16⋊3D5
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, C2×D8, C22×D5, C4○D16, D4×D5, D5×D8, D16⋊3D5
►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)
(1 144)(2 143)(3 142)(4 141)(5 140)(6 139)(7 138)(8 137)(9 136)(10 135)(11 134)(12 133)(13 132)(14 131)(15 130)(16 129)(17 49)(18 64)(19 63)(20 62)(21 61)(22 60)(23 59)(24 58)(25 57)(26 56)(27 55)(28 54)(29 53)(30 52)(31 51)(32 50)(33 77)(34 76)(35 75)(36 74)(37 73)(38 72)(39 71)(40 70)(41 69)(42 68)(43 67)(44 66)(45 65)(46 80)(47 79)(48 78)(81 158)(82 157)(83 156)(84 155)(85 154)(86 153)(87 152)(88 151)(89 150)(90 149)(91 148)(92 147)(93 146)(94 145)(95 160)(96 159)(97 123)(98 122)(99 121)(100 120)(101 119)(102 118)(103 117)(104 116)(105 115)(106 114)(107 113)(108 128)(109 127)(110 126)(111 125)(112 124)
(1 84 43 59 104)(2 85 44 60 105)(3 86 45 61 106)(4 87 46 62 107)(5 88 47 63 108)(6 89 48 64 109)(7 90 33 49 110)(8 91 34 50 111)(9 92 35 51 112)(10 93 36 52 97)(11 94 37 53 98)(12 95 38 54 99)(13 96 39 55 100)(14 81 40 56 101)(15 82 41 57 102)(16 83 42 58 103)(17 126 138 149 77)(18 127 139 150 78)(19 128 140 151 79)(20 113 141 152 80)(21 114 142 153 65)(22 115 143 154 66)(23 116 144 155 67)(24 117 129 156 68)(25 118 130 157 69)(26 119 131 158 70)(27 120 132 159 71)(28 121 133 160 72)(29 122 134 145 73)(30 123 135 146 74)(31 124 136 147 75)(32 125 137 148 76)
(1 104)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 111)(9 112)(10 97)(11 98)(12 99)(13 100)(14 101)(15 102)(16 103)(17 157)(18 158)(19 159)(20 160)(21 145)(22 146)(23 147)(24 148)(25 149)(26 150)(27 151)(28 152)(29 153)(30 154)(31 155)(32 156)(49 90)(50 91)(51 92)(52 93)(53 94)(54 95)(55 96)(56 81)(57 82)(58 83)(59 84)(60 85)(61 86)(62 87)(63 88)(64 89)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 80)(113 133)(114 134)(115 135)(116 136)(117 137)(118 138)(119 139)(120 140)(121 141)(122 142)(123 143)(124 144)(125 129)(126 130)(127 131)(128 132)
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), (1,144)(2,143)(3,142)(4,141)(5,140)(6,139)(7,138)(8,137)(9,136)(10,135)(11,134)(12,133)(13,132)(14,131)(15,130)(16,129)(17,49)(18,64)(19,63)(20,62)(21,61)(22,60)(23,59)(24,58)(25,57)(26,56)(27,55)(28,54)(29,53)(30,52)(31,51)(32,50)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,70)(41,69)(42,68)(43,67)(44,66)(45,65)(46,80)(47,79)(48,78)(81,158)(82,157)(83,156)(84,155)(85,154)(86,153)(87,152)(88,151)(89,150)(90,149)(91,148)(92,147)(93,146)(94,145)(95,160)(96,159)(97,123)(98,122)(99,121)(100,120)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,128)(109,127)(110,126)(111,125)(112,124), (1,84,43,59,104)(2,85,44,60,105)(3,86,45,61,106)(4,87,46,62,107)(5,88,47,63,108)(6,89,48,64,109)(7,90,33,49,110)(8,91,34,50,111)(9,92,35,51,112)(10,93,36,52,97)(11,94,37,53,98)(12,95,38,54,99)(13,96,39,55,100)(14,81,40,56,101)(15,82,41,57,102)(16,83,42,58,103)(17,126,138,149,77)(18,127,139,150,78)(19,128,140,151,79)(20,113,141,152,80)(21,114,142,153,65)(22,115,143,154,66)(23,116,144,155,67)(24,117,129,156,68)(25,118,130,157,69)(26,119,131,158,70)(27,120,132,159,71)(28,121,133,160,72)(29,122,134,145,73)(30,123,135,146,74)(31,124,136,147,75)(32,125,137,148,76), (1,104)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,111)(9,112)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,157)(18,158)(19,159)(20,160)(21,145)(22,146)(23,147)(24,148)(25,149)(26,150)(27,151)(28,152)(29,153)(30,154)(31,155)(32,156)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,81)(57,82)(58,83)(59,84)(60,85)(61,86)(62,87)(63,88)(64,89)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80)(113,133)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,140)(121,141)(122,142)(123,143)(124,144)(125,129)(126,130)(127,131)(128,132)>;
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), (1,144)(2,143)(3,142)(4,141)(5,140)(6,139)(7,138)(8,137)(9,136)(10,135)(11,134)(12,133)(13,132)(14,131)(15,130)(16,129)(17,49)(18,64)(19,63)(20,62)(21,61)(22,60)(23,59)(24,58)(25,57)(26,56)(27,55)(28,54)(29,53)(30,52)(31,51)(32,50)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,70)(41,69)(42,68)(43,67)(44,66)(45,65)(46,80)(47,79)(48,78)(81,158)(82,157)(83,156)(84,155)(85,154)(86,153)(87,152)(88,151)(89,150)(90,149)(91,148)(92,147)(93,146)(94,145)(95,160)(96,159)(97,123)(98,122)(99,121)(100,120)(101,119)(102,118)(103,117)(104,116)(105,115)(106,114)(107,113)(108,128)(109,127)(110,126)(111,125)(112,124), (1,84,43,59,104)(2,85,44,60,105)(3,86,45,61,106)(4,87,46,62,107)(5,88,47,63,108)(6,89,48,64,109)(7,90,33,49,110)(8,91,34,50,111)(9,92,35,51,112)(10,93,36,52,97)(11,94,37,53,98)(12,95,38,54,99)(13,96,39,55,100)(14,81,40,56,101)(15,82,41,57,102)(16,83,42,58,103)(17,126,138,149,77)(18,127,139,150,78)(19,128,140,151,79)(20,113,141,152,80)(21,114,142,153,65)(22,115,143,154,66)(23,116,144,155,67)(24,117,129,156,68)(25,118,130,157,69)(26,119,131,158,70)(27,120,132,159,71)(28,121,133,160,72)(29,122,134,145,73)(30,123,135,146,74)(31,124,136,147,75)(32,125,137,148,76), (1,104)(2,105)(3,106)(4,107)(5,108)(6,109)(7,110)(8,111)(9,112)(10,97)(11,98)(12,99)(13,100)(14,101)(15,102)(16,103)(17,157)(18,158)(19,159)(20,160)(21,145)(22,146)(23,147)(24,148)(25,149)(26,150)(27,151)(28,152)(29,153)(30,154)(31,155)(32,156)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,81)(57,82)(58,83)(59,84)(60,85)(61,86)(62,87)(63,88)(64,89)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80)(113,133)(114,134)(115,135)(116,136)(117,137)(118,138)(119,139)(120,140)(121,141)(122,142)(123,143)(124,144)(125,129)(126,130)(127,131)(128,132) );
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)], [(1,144),(2,143),(3,142),(4,141),(5,140),(6,139),(7,138),(8,137),(9,136),(10,135),(11,134),(12,133),(13,132),(14,131),(15,130),(16,129),(17,49),(18,64),(19,63),(20,62),(21,61),(22,60),(23,59),(24,58),(25,57),(26,56),(27,55),(28,54),(29,53),(30,52),(31,51),(32,50),(33,77),(34,76),(35,75),(36,74),(37,73),(38,72),(39,71),(40,70),(41,69),(42,68),(43,67),(44,66),(45,65),(46,80),(47,79),(48,78),(81,158),(82,157),(83,156),(84,155),(85,154),(86,153),(87,152),(88,151),(89,150),(90,149),(91,148),(92,147),(93,146),(94,145),(95,160),(96,159),(97,123),(98,122),(99,121),(100,120),(101,119),(102,118),(103,117),(104,116),(105,115),(106,114),(107,113),(108,128),(109,127),(110,126),(111,125),(112,124)], [(1,84,43,59,104),(2,85,44,60,105),(3,86,45,61,106),(4,87,46,62,107),(5,88,47,63,108),(6,89,48,64,109),(7,90,33,49,110),(8,91,34,50,111),(9,92,35,51,112),(10,93,36,52,97),(11,94,37,53,98),(12,95,38,54,99),(13,96,39,55,100),(14,81,40,56,101),(15,82,41,57,102),(16,83,42,58,103),(17,126,138,149,77),(18,127,139,150,78),(19,128,140,151,79),(20,113,141,152,80),(21,114,142,153,65),(22,115,143,154,66),(23,116,144,155,67),(24,117,129,156,68),(25,118,130,157,69),(26,119,131,158,70),(27,120,132,159,71),(28,121,133,160,72),(29,122,134,145,73),(30,123,135,146,74),(31,124,136,147,75),(32,125,137,148,76)], [(1,104),(2,105),(3,106),(4,107),(5,108),(6,109),(7,110),(8,111),(9,112),(10,97),(11,98),(12,99),(13,100),(14,101),(15,102),(16,103),(17,157),(18,158),(19,159),(20,160),(21,145),(22,146),(23,147),(24,148),(25,149),(26,150),(27,151),(28,152),(29,153),(30,154),(31,155),(32,156),(49,90),(50,91),(51,92),(52,93),(53,94),(54,95),(55,96),(56,81),(57,82),(58,83),(59,84),(60,85),(61,86),(62,87),(63,88),(64,89),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78),(71,79),(72,80),(113,133),(114,134),(115,135),(116,136),(117,137),(118,138),(119,139),(120,140),(121,141),(122,142),(123,143),(124,144),(125,129),(126,130),(127,131),(128,132)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | 40A | 40B | 40C | 40D | 80A | ··· | 80H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 8 | 8 | 10 | 2 | 5 | 5 | 40 | 40 | 2 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 16 | 16 | 16 | 16 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | D8 | D10 | D10 | C4○D16 | D4×D5 | D5×D8 | D16⋊3D5 |
| kernel | D16⋊3D5 | D5×C16 | Dic40 | D8.D5 | C5×D16 | D8⋊3D5 | C5⋊2C8 | C4×D5 | D16 | Dic5 | D10 | C16 | D8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 2 | 4 | 8 |
Matrix representation of D16⋊3D5 ►in GL4(𝔽241) generated by
| 165 | 0 | 0 | 0 |
| 0 | 130 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 130 | 0 | 0 |
| 165 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 1 |
| 0 | 0 | 50 | 190 |
| 1 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 50 | 1 |
G:=sub<GL(4,GF(241))| [165,0,0,0,0,130,0,0,0,0,240,0,0,0,0,240],[0,165,0,0,130,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,240,50,0,0,1,190],[1,0,0,0,0,240,0,0,0,0,240,50,0,0,0,1] >;
D16⋊3D5 in GAP, Magma, Sage, TeX
D_{16}\rtimes_3D_5 % in TeX
G:=Group("D16:3D5"); // GroupNames label
G:=SmallGroup(320,539);
// by ID
G=gap.SmallGroup(320,539);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,758,135,346,185,192,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^16=b^2=c^5=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^8*b,d*c*d=c^-1>;
// generators/relations