metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.D19, C4.2D38, C38.8D4, C19⋊2SD16, Dic38⋊2C2, C76.2C22, C19⋊C8⋊2C2, (D4×C19).1C2, C2.5(C19⋊D4), SmallGroup(304,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D19
G = < a,b,c,d | a4=b2=c19=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >
Smallest permutation representation of D4.D19
►On 152 points
Generators in S152
(1 67 24 56)(2 68 25 57)(3 69 26 39)(4 70 27 40)(5 71 28 41)(6 72 29 42)(7 73 30 43)(8 74 31 44)(9 75 32 45)(10 76 33 46)(11 58 34 47)(12 59 35 48)(13 60 36 49)(14 61 37 50)(15 62 38 51)(16 63 20 52)(17 64 21 53)(18 65 22 54)(19 66 23 55)(77 115 96 134)(78 116 97 135)(79 117 98 136)(80 118 99 137)(81 119 100 138)(82 120 101 139)(83 121 102 140)(84 122 103 141)(85 123 104 142)(86 124 105 143)(87 125 106 144)(88 126 107 145)(89 127 108 146)(90 128 109 147)(91 129 110 148)(92 130 111 149)(93 131 112 150)(94 132 113 151)(95 133 114 152)
(1 56)(2 57)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 63)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 71)(29 72)(30 73)(31 74)(32 75)(33 76)(34 58)(35 59)(36 60)(37 61)(38 62)(77 96)(78 97)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)(85 104)(86 105)(87 106)(88 107)(89 108)(90 109)(91 110)(92 111)(93 112)(94 113)(95 114)
(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)
(1 114 24 95)(2 113 25 94)(3 112 26 93)(4 111 27 92)(5 110 28 91)(6 109 29 90)(7 108 30 89)(8 107 31 88)(9 106 32 87)(10 105 33 86)(11 104 34 85)(12 103 35 84)(13 102 36 83)(14 101 37 82)(15 100 38 81)(16 99 20 80)(17 98 21 79)(18 97 22 78)(19 96 23 77)(39 150 69 131)(40 149 70 130)(41 148 71 129)(42 147 72 128)(43 146 73 127)(44 145 74 126)(45 144 75 125)(46 143 76 124)(47 142 58 123)(48 141 59 122)(49 140 60 121)(50 139 61 120)(51 138 62 119)(52 137 63 118)(53 136 64 117)(54 135 65 116)(55 134 66 115)(56 152 67 133)(57 151 68 132)
G:=sub<Sym(152)| (1,67,24,56)(2,68,25,57)(3,69,26,39)(4,70,27,40)(5,71,28,41)(6,72,29,42)(7,73,30,43)(8,74,31,44)(9,75,32,45)(10,76,33,46)(11,58,34,47)(12,59,35,48)(13,60,36,49)(14,61,37,50)(15,62,38,51)(16,63,20,52)(17,64,21,53)(18,65,22,54)(19,66,23,55)(77,115,96,134)(78,116,97,135)(79,117,98,136)(80,118,99,137)(81,119,100,138)(82,120,101,139)(83,121,102,140)(84,122,103,141)(85,123,104,142)(86,124,105,143)(87,125,106,144)(88,126,107,145)(89,127,108,146)(90,128,109,147)(91,129,110,148)(92,130,111,149)(93,131,112,150)(94,132,113,151)(95,133,114,152), (1,56)(2,57)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,63)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(33,76)(34,58)(35,59)(36,60)(37,61)(38,62)(77,96)(78,97)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103)(85,104)(86,105)(87,106)(88,107)(89,108)(90,109)(91,110)(92,111)(93,112)(94,113)(95,114), (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), (1,114,24,95)(2,113,25,94)(3,112,26,93)(4,111,27,92)(5,110,28,91)(6,109,29,90)(7,108,30,89)(8,107,31,88)(9,106,32,87)(10,105,33,86)(11,104,34,85)(12,103,35,84)(13,102,36,83)(14,101,37,82)(15,100,38,81)(16,99,20,80)(17,98,21,79)(18,97,22,78)(19,96,23,77)(39,150,69,131)(40,149,70,130)(41,148,71,129)(42,147,72,128)(43,146,73,127)(44,145,74,126)(45,144,75,125)(46,143,76,124)(47,142,58,123)(48,141,59,122)(49,140,60,121)(50,139,61,120)(51,138,62,119)(52,137,63,118)(53,136,64,117)(54,135,65,116)(55,134,66,115)(56,152,67,133)(57,151,68,132)>;
G:=Group( (1,67,24,56)(2,68,25,57)(3,69,26,39)(4,70,27,40)(5,71,28,41)(6,72,29,42)(7,73,30,43)(8,74,31,44)(9,75,32,45)(10,76,33,46)(11,58,34,47)(12,59,35,48)(13,60,36,49)(14,61,37,50)(15,62,38,51)(16,63,20,52)(17,64,21,53)(18,65,22,54)(19,66,23,55)(77,115,96,134)(78,116,97,135)(79,117,98,136)(80,118,99,137)(81,119,100,138)(82,120,101,139)(83,121,102,140)(84,122,103,141)(85,123,104,142)(86,124,105,143)(87,125,106,144)(88,126,107,145)(89,127,108,146)(90,128,109,147)(91,129,110,148)(92,130,111,149)(93,131,112,150)(94,132,113,151)(95,133,114,152), (1,56)(2,57)(3,39)(4,40)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,63)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,75)(33,76)(34,58)(35,59)(36,60)(37,61)(38,62)(77,96)(78,97)(79,98)(80,99)(81,100)(82,101)(83,102)(84,103)(85,104)(86,105)(87,106)(88,107)(89,108)(90,109)(91,110)(92,111)(93,112)(94,113)(95,114), (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), (1,114,24,95)(2,113,25,94)(3,112,26,93)(4,111,27,92)(5,110,28,91)(6,109,29,90)(7,108,30,89)(8,107,31,88)(9,106,32,87)(10,105,33,86)(11,104,34,85)(12,103,35,84)(13,102,36,83)(14,101,37,82)(15,100,38,81)(16,99,20,80)(17,98,21,79)(18,97,22,78)(19,96,23,77)(39,150,69,131)(40,149,70,130)(41,148,71,129)(42,147,72,128)(43,146,73,127)(44,145,74,126)(45,144,75,125)(46,143,76,124)(47,142,58,123)(48,141,59,122)(49,140,60,121)(50,139,61,120)(51,138,62,119)(52,137,63,118)(53,136,64,117)(54,135,65,116)(55,134,66,115)(56,152,67,133)(57,151,68,132) );
G=PermutationGroup([[(1,67,24,56),(2,68,25,57),(3,69,26,39),(4,70,27,40),(5,71,28,41),(6,72,29,42),(7,73,30,43),(8,74,31,44),(9,75,32,45),(10,76,33,46),(11,58,34,47),(12,59,35,48),(13,60,36,49),(14,61,37,50),(15,62,38,51),(16,63,20,52),(17,64,21,53),(18,65,22,54),(19,66,23,55),(77,115,96,134),(78,116,97,135),(79,117,98,136),(80,118,99,137),(81,119,100,138),(82,120,101,139),(83,121,102,140),(84,122,103,141),(85,123,104,142),(86,124,105,143),(87,125,106,144),(88,126,107,145),(89,127,108,146),(90,128,109,147),(91,129,110,148),(92,130,111,149),(93,131,112,150),(94,132,113,151),(95,133,114,152)], [(1,56),(2,57),(3,39),(4,40),(5,41),(6,42),(7,43),(8,44),(9,45),(10,46),(11,47),(12,48),(13,49),(14,50),(15,51),(16,52),(17,53),(18,54),(19,55),(20,63),(21,64),(22,65),(23,66),(24,67),(25,68),(26,69),(27,70),(28,71),(29,72),(30,73),(31,74),(32,75),(33,76),(34,58),(35,59),(36,60),(37,61),(38,62),(77,96),(78,97),(79,98),(80,99),(81,100),(82,101),(83,102),(84,103),(85,104),(86,105),(87,106),(88,107),(89,108),(90,109),(91,110),(92,111),(93,112),(94,113),(95,114)], [(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)], [(1,114,24,95),(2,113,25,94),(3,112,26,93),(4,111,27,92),(5,110,28,91),(6,109,29,90),(7,108,30,89),(8,107,31,88),(9,106,32,87),(10,105,33,86),(11,104,34,85),(12,103,35,84),(13,102,36,83),(14,101,37,82),(15,100,38,81),(16,99,20,80),(17,98,21,79),(18,97,22,78),(19,96,23,77),(39,150,69,131),(40,149,70,130),(41,148,71,129),(42,147,72,128),(43,146,73,127),(44,145,74,126),(45,144,75,125),(46,143,76,124),(47,142,58,123),(48,141,59,122),(49,140,60,121),(50,139,61,120),(51,138,62,119),(52,137,63,118),(53,136,64,117),(54,135,65,116),(55,134,66,115),(56,152,67,133),(57,151,68,132)]])
52 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 19A | ··· | 19I | 38A | ··· | 38I | 38J | ··· | 38AA | 76A | ··· | 76I |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 19 | ··· | 19 | 38 | ··· | 38 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 4 | 2 | 76 | 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 | SD16 | D19 | D38 | C19⋊D4 | D4.D19 |
| kernel | D4.D19 | C19⋊C8 | Dic38 | 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 | 203 | 327 |
| 0 | 0 | 317 | 254 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 203 | 327 |
| 0 | 0 | 92 | 254 |
| 381 | 1 | 0 | 0 |
| 452 | 433 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 422 | 394 | 0 | 0 |
| 150 | 35 | 0 | 0 |
| 0 | 0 | 81 | 337 |
| 0 | 0 | 108 | 376 |
G:=sub<GL(4,GF(457))| [1,0,0,0,0,1,0,0,0,0,203,317,0,0,327,254],[1,0,0,0,0,1,0,0,0,0,203,92,0,0,327,254],[381,452,0,0,1,433,0,0,0,0,1,0,0,0,0,1],[422,150,0,0,394,35,0,0,0,0,81,108,0,0,337,376] >;
D4.D19 in GAP, Magma, Sage, TeX
D_4.D_{19} % in TeX
G:=Group("D4.D19"); // GroupNames label
G:=SmallGroup(304,15);
// by ID
G=gap.SmallGroup(304,15);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,40,61,182,97,42,7204]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^19=1,d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations
Export