metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊D19, C4.3D38, C38.9D4, C19⋊3SD16, D76.2C2, C76.3C22, C19⋊C8⋊3C2, (Q8×C19)⋊1C2, C2.6(C19⋊D4), SmallGroup(304,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊D19
G = < a,b,c,d | a4=c19=d2=1, b2=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a-1b, dcd=c-1 >
Smallest permutation representation of Q8⋊D19
►On 152 points
Generators in S152
(1 70 31 52)(2 71 32 53)(3 72 33 54)(4 73 34 55)(5 74 35 56)(6 75 36 57)(7 76 37 39)(8 58 38 40)(9 59 20 41)(10 60 21 42)(11 61 22 43)(12 62 23 44)(13 63 24 45)(14 64 25 46)(15 65 26 47)(16 66 27 48)(17 67 28 49)(18 68 29 50)(19 69 30 51)(77 130 105 149)(78 131 106 150)(79 132 107 151)(80 133 108 152)(81 115 109 134)(82 116 110 135)(83 117 111 136)(84 118 112 137)(85 119 113 138)(86 120 114 139)(87 121 96 140)(88 122 97 141)(89 123 98 142)(90 124 99 143)(91 125 100 144)(92 126 101 145)(93 127 102 146)(94 128 103 147)(95 129 104 148)
(1 112 31 84)(2 113 32 85)(3 114 33 86)(4 96 34 87)(5 97 35 88)(6 98 36 89)(7 99 37 90)(8 100 38 91)(9 101 20 92)(10 102 21 93)(11 103 22 94)(12 104 23 95)(13 105 24 77)(14 106 25 78)(15 107 26 79)(16 108 27 80)(17 109 28 81)(18 110 29 82)(19 111 30 83)(39 143 76 124)(40 144 58 125)(41 145 59 126)(42 146 60 127)(43 147 61 128)(44 148 62 129)(45 149 63 130)(46 150 64 131)(47 151 65 132)(48 152 66 133)(49 134 67 115)(50 135 68 116)(51 136 69 117)(52 137 70 118)(53 138 71 119)(54 139 72 120)(55 140 73 121)(56 141 74 122)(57 142 75 123)
(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 22)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(39 63)(40 62)(41 61)(42 60)(43 59)(44 58)(45 76)(46 75)(47 74)(48 73)(49 72)(50 71)(51 70)(52 69)(53 68)(54 67)(55 66)(56 65)(57 64)(77 124)(78 123)(79 122)(80 121)(81 120)(82 119)(83 118)(84 117)(85 116)(86 115)(87 133)(88 132)(89 131)(90 130)(91 129)(92 128)(93 127)(94 126)(95 125)(96 152)(97 151)(98 150)(99 149)(100 148)(101 147)(102 146)(103 145)(104 144)(105 143)(106 142)(107 141)(108 140)(109 139)(110 138)(111 137)(112 136)(113 135)(114 134)
G:=sub<Sym(152)| (1,70,31,52)(2,71,32,53)(3,72,33,54)(4,73,34,55)(5,74,35,56)(6,75,36,57)(7,76,37,39)(8,58,38,40)(9,59,20,41)(10,60,21,42)(11,61,22,43)(12,62,23,44)(13,63,24,45)(14,64,25,46)(15,65,26,47)(16,66,27,48)(17,67,28,49)(18,68,29,50)(19,69,30,51)(77,130,105,149)(78,131,106,150)(79,132,107,151)(80,133,108,152)(81,115,109,134)(82,116,110,135)(83,117,111,136)(84,118,112,137)(85,119,113,138)(86,120,114,139)(87,121,96,140)(88,122,97,141)(89,123,98,142)(90,124,99,143)(91,125,100,144)(92,126,101,145)(93,127,102,146)(94,128,103,147)(95,129,104,148), (1,112,31,84)(2,113,32,85)(3,114,33,86)(4,96,34,87)(5,97,35,88)(6,98,36,89)(7,99,37,90)(8,100,38,91)(9,101,20,92)(10,102,21,93)(11,103,22,94)(12,104,23,95)(13,105,24,77)(14,106,25,78)(15,107,26,79)(16,108,27,80)(17,109,28,81)(18,110,29,82)(19,111,30,83)(39,143,76,124)(40,144,58,125)(41,145,59,126)(42,146,60,127)(43,147,61,128)(44,148,62,129)(45,149,63,130)(46,150,64,131)(47,151,65,132)(48,152,66,133)(49,134,67,115)(50,135,68,116)(51,136,69,117)(52,137,70,118)(53,138,71,119)(54,139,72,120)(55,140,73,121)(56,141,74,122)(57,142,75,123), (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)(20,22)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(39,63)(40,62)(41,61)(42,60)(43,59)(44,58)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(77,124)(78,123)(79,122)(80,121)(81,120)(82,119)(83,118)(84,117)(85,116)(86,115)(87,133)(88,132)(89,131)(90,130)(91,129)(92,128)(93,127)(94,126)(95,125)(96,152)(97,151)(98,150)(99,149)(100,148)(101,147)(102,146)(103,145)(104,144)(105,143)(106,142)(107,141)(108,140)(109,139)(110,138)(111,137)(112,136)(113,135)(114,134)>;
G:=Group( (1,70,31,52)(2,71,32,53)(3,72,33,54)(4,73,34,55)(5,74,35,56)(6,75,36,57)(7,76,37,39)(8,58,38,40)(9,59,20,41)(10,60,21,42)(11,61,22,43)(12,62,23,44)(13,63,24,45)(14,64,25,46)(15,65,26,47)(16,66,27,48)(17,67,28,49)(18,68,29,50)(19,69,30,51)(77,130,105,149)(78,131,106,150)(79,132,107,151)(80,133,108,152)(81,115,109,134)(82,116,110,135)(83,117,111,136)(84,118,112,137)(85,119,113,138)(86,120,114,139)(87,121,96,140)(88,122,97,141)(89,123,98,142)(90,124,99,143)(91,125,100,144)(92,126,101,145)(93,127,102,146)(94,128,103,147)(95,129,104,148), (1,112,31,84)(2,113,32,85)(3,114,33,86)(4,96,34,87)(5,97,35,88)(6,98,36,89)(7,99,37,90)(8,100,38,91)(9,101,20,92)(10,102,21,93)(11,103,22,94)(12,104,23,95)(13,105,24,77)(14,106,25,78)(15,107,26,79)(16,108,27,80)(17,109,28,81)(18,110,29,82)(19,111,30,83)(39,143,76,124)(40,144,58,125)(41,145,59,126)(42,146,60,127)(43,147,61,128)(44,148,62,129)(45,149,63,130)(46,150,64,131)(47,151,65,132)(48,152,66,133)(49,134,67,115)(50,135,68,116)(51,136,69,117)(52,137,70,118)(53,138,71,119)(54,139,72,120)(55,140,73,121)(56,141,74,122)(57,142,75,123), (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)(20,22)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(39,63)(40,62)(41,61)(42,60)(43,59)(44,58)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(77,124)(78,123)(79,122)(80,121)(81,120)(82,119)(83,118)(84,117)(85,116)(86,115)(87,133)(88,132)(89,131)(90,130)(91,129)(92,128)(93,127)(94,126)(95,125)(96,152)(97,151)(98,150)(99,149)(100,148)(101,147)(102,146)(103,145)(104,144)(105,143)(106,142)(107,141)(108,140)(109,139)(110,138)(111,137)(112,136)(113,135)(114,134) );
G=PermutationGroup([[(1,70,31,52),(2,71,32,53),(3,72,33,54),(4,73,34,55),(5,74,35,56),(6,75,36,57),(7,76,37,39),(8,58,38,40),(9,59,20,41),(10,60,21,42),(11,61,22,43),(12,62,23,44),(13,63,24,45),(14,64,25,46),(15,65,26,47),(16,66,27,48),(17,67,28,49),(18,68,29,50),(19,69,30,51),(77,130,105,149),(78,131,106,150),(79,132,107,151),(80,133,108,152),(81,115,109,134),(82,116,110,135),(83,117,111,136),(84,118,112,137),(85,119,113,138),(86,120,114,139),(87,121,96,140),(88,122,97,141),(89,123,98,142),(90,124,99,143),(91,125,100,144),(92,126,101,145),(93,127,102,146),(94,128,103,147),(95,129,104,148)], [(1,112,31,84),(2,113,32,85),(3,114,33,86),(4,96,34,87),(5,97,35,88),(6,98,36,89),(7,99,37,90),(8,100,38,91),(9,101,20,92),(10,102,21,93),(11,103,22,94),(12,104,23,95),(13,105,24,77),(14,106,25,78),(15,107,26,79),(16,108,27,80),(17,109,28,81),(18,110,29,82),(19,111,30,83),(39,143,76,124),(40,144,58,125),(41,145,59,126),(42,146,60,127),(43,147,61,128),(44,148,62,129),(45,149,63,130),(46,150,64,131),(47,151,65,132),(48,152,66,133),(49,134,67,115),(50,135,68,116),(51,136,69,117),(52,137,70,118),(53,138,71,119),(54,139,72,120),(55,140,73,121),(56,141,74,122),(57,142,75,123)], [(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),(20,22),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(39,63),(40,62),(41,61),(42,60),(43,59),(44,58),(45,76),(46,75),(47,74),(48,73),(49,72),(50,71),(51,70),(52,69),(53,68),(54,67),(55,66),(56,65),(57,64),(77,124),(78,123),(79,122),(80,121),(81,120),(82,119),(83,118),(84,117),(85,116),(86,115),(87,133),(88,132),(89,131),(90,130),(91,129),(92,128),(93,127),(94,126),(95,125),(96,152),(97,151),(98,150),(99,149),(100,148),(101,147),(102,146),(103,145),(104,144),(105,143),(106,142),(107,141),(108,140),(109,139),(110,138),(111,137),(112,136),(113,135),(114,134)]])
52 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 19A | ··· | 19I | 38A | ··· | 38I | 76A | ··· | 76AA |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 76 | 2 | 4 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 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 | Q8⋊D19 |
| kernel | Q8⋊D19 | C19⋊C8 | D76 | Q8×C19 | C38 | C19 | Q8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 9 | 9 | 18 | 9 |
Matrix representation of Q8⋊D19 ►in GL4(𝔽457) generated by
| 0 | 1 | 0 | 0 |
| 456 | 0 | 0 | 0 |
| 0 | 0 | 456 | 0 |
| 0 | 0 | 0 | 456 |
| 247 | 247 | 0 | 0 |
| 247 | 210 | 0 | 0 |
| 0 | 0 | 99 | 270 |
| 0 | 0 | 353 | 358 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 303 | 1 |
| 0 | 0 | 3 | 89 |
| 1 | 0 | 0 | 0 |
| 0 | 456 | 0 | 0 |
| 0 | 0 | 293 | 62 |
| 0 | 0 | 178 | 164 |
G:=sub<GL(4,GF(457))| [0,456,0,0,1,0,0,0,0,0,456,0,0,0,0,456],[247,247,0,0,247,210,0,0,0,0,99,353,0,0,270,358],[1,0,0,0,0,1,0,0,0,0,303,3,0,0,1,89],[1,0,0,0,0,456,0,0,0,0,293,178,0,0,62,164] >;
Q8⋊D19 in GAP, Magma, Sage, TeX
Q_8\rtimes D_{19} % in TeX
G:=Group("Q8:D19"); // GroupNames label
G:=SmallGroup(304,16);
// by ID
G=gap.SmallGroup(304,16);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,61,46,182,97,42,7204]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^19=d^2=1,b^2=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^-1*b,d*c*d=c^-1>;
// generators/relations
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