direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: D8×C19, D4⋊C38, C8⋊1C38, C152⋊5C2, C38.14D4, C76.17C22, (D4×C19)⋊4C2, C4.1(C2×C38), C2.3(D4×C19), SmallGroup(304,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D8×C19
G = < a,b,c | a19=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D8×C19
►On 152 points
Generators in S152
(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 38 60 121 112 79 143 39)(2 20 61 122 113 80 144 40)(3 21 62 123 114 81 145 41)(4 22 63 124 96 82 146 42)(5 23 64 125 97 83 147 43)(6 24 65 126 98 84 148 44)(7 25 66 127 99 85 149 45)(8 26 67 128 100 86 150 46)(9 27 68 129 101 87 151 47)(10 28 69 130 102 88 152 48)(11 29 70 131 103 89 134 49)(12 30 71 132 104 90 135 50)(13 31 72 133 105 91 136 51)(14 32 73 115 106 92 137 52)(15 33 74 116 107 93 138 53)(16 34 75 117 108 94 139 54)(17 35 76 118 109 95 140 55)(18 36 58 119 110 77 141 56)(19 37 59 120 111 78 142 57)
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 55)(18 56)(19 57)(20 144)(21 145)(22 146)(23 147)(24 148)(25 149)(26 150)(27 151)(28 152)(29 134)(30 135)(31 136)(32 137)(33 138)(34 139)(35 140)(36 141)(37 142)(38 143)(58 77)(59 78)(60 79)(61 80)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)(73 92)(74 93)(75 94)(76 95)(96 124)(97 125)(98 126)(99 127)(100 128)(101 129)(102 130)(103 131)(104 132)(105 133)(106 115)(107 116)(108 117)(109 118)(110 119)(111 120)(112 121)(113 122)(114 123)
G:=sub<Sym(152)| (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), (1,38,60,121,112,79,143,39)(2,20,61,122,113,80,144,40)(3,21,62,123,114,81,145,41)(4,22,63,124,96,82,146,42)(5,23,64,125,97,83,147,43)(6,24,65,126,98,84,148,44)(7,25,66,127,99,85,149,45)(8,26,67,128,100,86,150,46)(9,27,68,129,101,87,151,47)(10,28,69,130,102,88,152,48)(11,29,70,131,103,89,134,49)(12,30,71,132,104,90,135,50)(13,31,72,133,105,91,136,51)(14,32,73,115,106,92,137,52)(15,33,74,116,107,93,138,53)(16,34,75,117,108,94,139,54)(17,35,76,118,109,95,140,55)(18,36,58,119,110,77,141,56)(19,37,59,120,111,78,142,57), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,144)(21,145)(22,146)(23,147)(24,148)(25,149)(26,150)(27,151)(28,152)(29,134)(30,135)(31,136)(32,137)(33,138)(34,139)(35,140)(36,141)(37,142)(38,143)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(96,124)(97,125)(98,126)(99,127)(100,128)(101,129)(102,130)(103,131)(104,132)(105,133)(106,115)(107,116)(108,117)(109,118)(110,119)(111,120)(112,121)(113,122)(114,123)>;
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), (1,38,60,121,112,79,143,39)(2,20,61,122,113,80,144,40)(3,21,62,123,114,81,145,41)(4,22,63,124,96,82,146,42)(5,23,64,125,97,83,147,43)(6,24,65,126,98,84,148,44)(7,25,66,127,99,85,149,45)(8,26,67,128,100,86,150,46)(9,27,68,129,101,87,151,47)(10,28,69,130,102,88,152,48)(11,29,70,131,103,89,134,49)(12,30,71,132,104,90,135,50)(13,31,72,133,105,91,136,51)(14,32,73,115,106,92,137,52)(15,33,74,116,107,93,138,53)(16,34,75,117,108,94,139,54)(17,35,76,118,109,95,140,55)(18,36,58,119,110,77,141,56)(19,37,59,120,111,78,142,57), (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,55)(18,56)(19,57)(20,144)(21,145)(22,146)(23,147)(24,148)(25,149)(26,150)(27,151)(28,152)(29,134)(30,135)(31,136)(32,137)(33,138)(34,139)(35,140)(36,141)(37,142)(38,143)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91)(73,92)(74,93)(75,94)(76,95)(96,124)(97,125)(98,126)(99,127)(100,128)(101,129)(102,130)(103,131)(104,132)(105,133)(106,115)(107,116)(108,117)(109,118)(110,119)(111,120)(112,121)(113,122)(114,123) );
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)], [(1,38,60,121,112,79,143,39),(2,20,61,122,113,80,144,40),(3,21,62,123,114,81,145,41),(4,22,63,124,96,82,146,42),(5,23,64,125,97,83,147,43),(6,24,65,126,98,84,148,44),(7,25,66,127,99,85,149,45),(8,26,67,128,100,86,150,46),(9,27,68,129,101,87,151,47),(10,28,69,130,102,88,152,48),(11,29,70,131,103,89,134,49),(12,30,71,132,104,90,135,50),(13,31,72,133,105,91,136,51),(14,32,73,115,106,92,137,52),(15,33,74,116,107,93,138,53),(16,34,75,117,108,94,139,54),(17,35,76,118,109,95,140,55),(18,36,58,119,110,77,141,56),(19,37,59,120,111,78,142,57)], [(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,55),(18,56),(19,57),(20,144),(21,145),(22,146),(23,147),(24,148),(25,149),(26,150),(27,151),(28,152),(29,134),(30,135),(31,136),(32,137),(33,138),(34,139),(35,140),(36,141),(37,142),(38,143),(58,77),(59,78),(60,79),(61,80),(62,81),(63,82),(64,83),(65,84),(66,85),(67,86),(68,87),(69,88),(70,89),(71,90),(72,91),(73,92),(74,93),(75,94),(76,95),(96,124),(97,125),(98,126),(99,127),(100,128),(101,129),(102,130),(103,131),(104,132),(105,133),(106,115),(107,116),(108,117),(109,118),(110,119),(111,120),(112,121),(113,122),(114,123)]])
133 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 8A | 8B | 19A | ··· | 19R | 38A | ··· | 38R | 38S | ··· | 38BB | 76A | ··· | 76R | 152A | ··· | 152AJ |
| order | 1 | 2 | 2 | 2 | 4 | 8 | 8 | 19 | ··· | 19 | 38 | ··· | 38 | 38 | ··· | 38 | 76 | ··· | 76 | 152 | ··· | 152 |
| size | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 2 | ··· | 2 |
133 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C19 | C38 | C38 | D4 | D8 | D4×C19 | D8×C19 |
| kernel | D8×C19 | C152 | D4×C19 | D8 | C8 | D4 | C38 | C19 | C2 | C1 |
| # reps | 1 | 1 | 2 | 18 | 18 | 36 | 1 | 2 | 18 | 36 |
Matrix representation of D8×C19 ►in GL2(𝔽457) generated by
| 241 | 0 |
| 0 | 241 |
| 377 | 40 |
| 377 | 0 |
| 377 | 40 |
| 377 | 80 |
G:=sub<GL(2,GF(457))| [241,0,0,241],[377,377,40,0],[377,377,40,80] >;
D8×C19 in GAP, Magma, Sage, TeX
D_8\times C_{19} % in TeX
G:=Group("D8xC19"); // GroupNames label
G:=SmallGroup(304,24);
// by ID
G=gap.SmallGroup(304,24);
# by ID
G:=PCGroup([5,-2,-2,-19,-2,-2,781,4563,2288,58]);
// Polycyclic
G:=Group<a,b,c|a^19=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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