metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊D19, C19⋊2D8, D76⋊2C2, C4.1D38, C38.7D4, C76.1C22, C19⋊C8⋊1C2, (D4×C19)⋊1C2, C2.4(C19⋊D4), SmallGroup(304,14)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊D19
G = < a,b,c,d | a4=b2=c19=d2=1, bab=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=c-1 >
Smallest permutation representation of D4⋊D19
►On 152 points
Generators in S152
(1 64 32 44)(2 65 33 45)(3 66 34 46)(4 67 35 47)(5 68 36 48)(6 69 37 49)(7 70 38 50)(8 71 20 51)(9 72 21 52)(10 73 22 53)(11 74 23 54)(12 75 24 55)(13 76 25 56)(14 58 26 57)(15 59 27 39)(16 60 28 40)(17 61 29 41)(18 62 30 42)(19 63 31 43)(77 128 106 140)(78 129 107 141)(79 130 108 142)(80 131 109 143)(81 132 110 144)(82 133 111 145)(83 115 112 146)(84 116 113 147)(85 117 114 148)(86 118 96 149)(87 119 97 150)(88 120 98 151)(89 121 99 152)(90 122 100 134)(91 123 101 135)(92 124 102 136)(93 125 103 137)(94 126 104 138)(95 127 105 139)
(1 139)(2 140)(3 141)(4 142)(5 143)(6 144)(7 145)(8 146)(9 147)(10 148)(11 149)(12 150)(13 151)(14 152)(15 134)(16 135)(17 136)(18 137)(19 138)(20 115)(21 116)(22 117)(23 118)(24 119)(25 120)(26 121)(27 122)(28 123)(29 124)(30 125)(31 126)(32 127)(33 128)(34 129)(35 130)(36 131)(37 132)(38 133)(39 90)(40 91)(41 92)(42 93)(43 94)(44 95)(45 77)(46 78)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)(53 85)(54 86)(55 87)(56 88)(57 89)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 113)(73 114)(74 96)(75 97)(76 98)
(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)
(1 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 24)(21 23)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(31 32)(39 68)(40 67)(41 66)(42 65)(43 64)(44 63)(45 62)(46 61)(47 60)(48 59)(49 58)(50 76)(51 75)(52 74)(53 73)(54 72)(55 71)(56 70)(57 69)(77 125)(78 124)(79 123)(80 122)(81 121)(82 120)(83 119)(84 118)(85 117)(86 116)(87 115)(88 133)(89 132)(90 131)(91 130)(92 129)(93 128)(94 127)(95 126)(96 147)(97 146)(98 145)(99 144)(100 143)(101 142)(102 141)(103 140)(104 139)(105 138)(106 137)(107 136)(108 135)(109 134)(110 152)(111 151)(112 150)(113 149)(114 148)
G:=sub<Sym(152)| (1,64,32,44)(2,65,33,45)(3,66,34,46)(4,67,35,47)(5,68,36,48)(6,69,37,49)(7,70,38,50)(8,71,20,51)(9,72,21,52)(10,73,22,53)(11,74,23,54)(12,75,24,55)(13,76,25,56)(14,58,26,57)(15,59,27,39)(16,60,28,40)(17,61,29,41)(18,62,30,42)(19,63,31,43)(77,128,106,140)(78,129,107,141)(79,130,108,142)(80,131,109,143)(81,132,110,144)(82,133,111,145)(83,115,112,146)(84,116,113,147)(85,117,114,148)(86,118,96,149)(87,119,97,150)(88,120,98,151)(89,121,99,152)(90,122,100,134)(91,123,101,135)(92,124,102,136)(93,125,103,137)(94,126,104,138)(95,127,105,139), (1,139)(2,140)(3,141)(4,142)(5,143)(6,144)(7,145)(8,146)(9,147)(10,148)(11,149)(12,150)(13,151)(14,152)(15,134)(16,135)(17,136)(18,137)(19,138)(20,115)(21,116)(22,117)(23,118)(24,119)(25,120)(26,121)(27,122)(28,123)(29,124)(30,125)(31,126)(32,127)(33,128)(34,129)(35,130)(36,131)(37,132)(38,133)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,89)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,113)(73,114)(74,96)(75,97)(76,98), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,24)(21,23)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,76)(51,75)(52,74)(53,73)(54,72)(55,71)(56,70)(57,69)(77,125)(78,124)(79,123)(80,122)(81,121)(82,120)(83,119)(84,118)(85,117)(86,116)(87,115)(88,133)(89,132)(90,131)(91,130)(92,129)(93,128)(94,127)(95,126)(96,147)(97,146)(98,145)(99,144)(100,143)(101,142)(102,141)(103,140)(104,139)(105,138)(106,137)(107,136)(108,135)(109,134)(110,152)(111,151)(112,150)(113,149)(114,148)>;
G:=Group( (1,64,32,44)(2,65,33,45)(3,66,34,46)(4,67,35,47)(5,68,36,48)(6,69,37,49)(7,70,38,50)(8,71,20,51)(9,72,21,52)(10,73,22,53)(11,74,23,54)(12,75,24,55)(13,76,25,56)(14,58,26,57)(15,59,27,39)(16,60,28,40)(17,61,29,41)(18,62,30,42)(19,63,31,43)(77,128,106,140)(78,129,107,141)(79,130,108,142)(80,131,109,143)(81,132,110,144)(82,133,111,145)(83,115,112,146)(84,116,113,147)(85,117,114,148)(86,118,96,149)(87,119,97,150)(88,120,98,151)(89,121,99,152)(90,122,100,134)(91,123,101,135)(92,124,102,136)(93,125,103,137)(94,126,104,138)(95,127,105,139), (1,139)(2,140)(3,141)(4,142)(5,143)(6,144)(7,145)(8,146)(9,147)(10,148)(11,149)(12,150)(13,151)(14,152)(15,134)(16,135)(17,136)(18,137)(19,138)(20,115)(21,116)(22,117)(23,118)(24,119)(25,120)(26,121)(27,122)(28,123)(29,124)(30,125)(31,126)(32,127)(33,128)(34,129)(35,130)(36,131)(37,132)(38,133)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,88)(57,89)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,113)(73,114)(74,96)(75,97)(76,98), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,24)(21,23)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,76)(51,75)(52,74)(53,73)(54,72)(55,71)(56,70)(57,69)(77,125)(78,124)(79,123)(80,122)(81,121)(82,120)(83,119)(84,118)(85,117)(86,116)(87,115)(88,133)(89,132)(90,131)(91,130)(92,129)(93,128)(94,127)(95,126)(96,147)(97,146)(98,145)(99,144)(100,143)(101,142)(102,141)(103,140)(104,139)(105,138)(106,137)(107,136)(108,135)(109,134)(110,152)(111,151)(112,150)(113,149)(114,148) );
G=PermutationGroup([[(1,64,32,44),(2,65,33,45),(3,66,34,46),(4,67,35,47),(5,68,36,48),(6,69,37,49),(7,70,38,50),(8,71,20,51),(9,72,21,52),(10,73,22,53),(11,74,23,54),(12,75,24,55),(13,76,25,56),(14,58,26,57),(15,59,27,39),(16,60,28,40),(17,61,29,41),(18,62,30,42),(19,63,31,43),(77,128,106,140),(78,129,107,141),(79,130,108,142),(80,131,109,143),(81,132,110,144),(82,133,111,145),(83,115,112,146),(84,116,113,147),(85,117,114,148),(86,118,96,149),(87,119,97,150),(88,120,98,151),(89,121,99,152),(90,122,100,134),(91,123,101,135),(92,124,102,136),(93,125,103,137),(94,126,104,138),(95,127,105,139)], [(1,139),(2,140),(3,141),(4,142),(5,143),(6,144),(7,145),(8,146),(9,147),(10,148),(11,149),(12,150),(13,151),(14,152),(15,134),(16,135),(17,136),(18,137),(19,138),(20,115),(21,116),(22,117),(23,118),(24,119),(25,120),(26,121),(27,122),(28,123),(29,124),(30,125),(31,126),(32,127),(33,128),(34,129),(35,130),(36,131),(37,132),(38,133),(39,90),(40,91),(41,92),(42,93),(43,94),(44,95),(45,77),(46,78),(47,79),(48,80),(49,81),(50,82),(51,83),(52,84),(53,85),(54,86),(55,87),(56,88),(57,89),(58,99),(59,100),(60,101),(61,102),(62,103),(63,104),(64,105),(65,106),(66,107),(67,108),(68,109),(69,110),(70,111),(71,112),(72,113),(73,114),(74,96),(75,97),(76,98)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(20,24),(21,23),(25,38),(26,37),(27,36),(28,35),(29,34),(30,33),(31,32),(39,68),(40,67),(41,66),(42,65),(43,64),(44,63),(45,62),(46,61),(47,60),(48,59),(49,58),(50,76),(51,75),(52,74),(53,73),(54,72),(55,71),(56,70),(57,69),(77,125),(78,124),(79,123),(80,122),(81,121),(82,120),(83,119),(84,118),(85,117),(86,116),(87,115),(88,133),(89,132),(90,131),(91,130),(92,129),(93,128),(94,127),(95,126),(96,147),(97,146),(98,145),(99,144),(100,143),(101,142),(102,141),(103,140),(104,139),(105,138),(106,137),(107,136),(108,135),(109,134),(110,152),(111,151),(112,150),(113,149),(114,148)]])
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 8A | 8B | 19A | ··· | 19I | 38A | ··· | 38I | 38J | ··· | 38AA | 76A | ··· | 76I |
| order | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 19 | ··· | 19 | 38 | ··· | 38 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 4 | 76 | 2 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D8 | D19 | D38 | C19⋊D4 | D4⋊D19 |
| kernel | D4⋊D19 | C19⋊C8 | D76 | D4×C19 | C38 | C19 | D4 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 9 | 9 | 18 | 9 |
Matrix representation of D4⋊D19 ►in GL4(𝔽457) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 150 |
| 0 | 0 | 262 | 456 |
| 456 | 0 | 0 | 0 |
| 0 | 456 | 0 | 0 |
| 0 | 0 | 0 | 59 |
| 0 | 0 | 31 | 0 |
| 340 | 1 | 0 | 0 |
| 174 | 61 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 455 | 441 | 0 | 0 |
| 143 | 2 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 262 | 456 |
G:=sub<GL(4,GF(457))| [1,0,0,0,0,1,0,0,0,0,1,262,0,0,150,456],[456,0,0,0,0,456,0,0,0,0,0,31,0,0,59,0],[340,174,0,0,1,61,0,0,0,0,1,0,0,0,0,1],[455,143,0,0,441,2,0,0,0,0,1,262,0,0,0,456] >;
D4⋊D19 in GAP, Magma, Sage, TeX
D_4\rtimes D_{19} % in TeX
G:=Group("D4:D19"); // GroupNames label
G:=SmallGroup(304,14);
// by ID
G=gap.SmallGroup(304,14);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,61,182,97,42,7204]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^19=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations
Export