direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: SD16×C19, Q8⋊C38, C8⋊2C38, D4.C38, C152⋊6C2, C38.15D4, C76.18C22, C4.2(C2×C38), (Q8×C19)⋊4C2, C2.4(D4×C19), (D4×C19).2C2, SmallGroup(304,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD16×C19
G = < a,b,c | a19=b8=c2=1, ab=ba, ac=ca, cbc=b3 >
Smallest permutation representation of SD16×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 42 25 105 66 95 140 127)(2 43 26 106 67 77 141 128)(3 44 27 107 68 78 142 129)(4 45 28 108 69 79 143 130)(5 46 29 109 70 80 144 131)(6 47 30 110 71 81 145 132)(7 48 31 111 72 82 146 133)(8 49 32 112 73 83 147 115)(9 50 33 113 74 84 148 116)(10 51 34 114 75 85 149 117)(11 52 35 96 76 86 150 118)(12 53 36 97 58 87 151 119)(13 54 37 98 59 88 152 120)(14 55 38 99 60 89 134 121)(15 56 20 100 61 90 135 122)(16 57 21 101 62 91 136 123)(17 39 22 102 63 92 137 124)(18 40 23 103 64 93 138 125)(19 41 24 104 65 94 139 126)
(20 135)(21 136)(22 137)(23 138)(24 139)(25 140)(26 141)(27 142)(28 143)(29 144)(30 145)(31 146)(32 147)(33 148)(34 149)(35 150)(36 151)(37 152)(38 134)(39 102)(40 103)(41 104)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 113)(51 114)(52 96)(53 97)(54 98)(55 99)(56 100)(57 101)(77 128)(78 129)(79 130)(80 131)(81 132)(82 133)(83 115)(84 116)(85 117)(86 118)(87 119)(88 120)(89 121)(90 122)(91 123)(92 124)(93 125)(94 126)(95 127)
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), (1,42,25,105,66,95,140,127)(2,43,26,106,67,77,141,128)(3,44,27,107,68,78,142,129)(4,45,28,108,69,79,143,130)(5,46,29,109,70,80,144,131)(6,47,30,110,71,81,145,132)(7,48,31,111,72,82,146,133)(8,49,32,112,73,83,147,115)(9,50,33,113,74,84,148,116)(10,51,34,114,75,85,149,117)(11,52,35,96,76,86,150,118)(12,53,36,97,58,87,151,119)(13,54,37,98,59,88,152,120)(14,55,38,99,60,89,134,121)(15,56,20,100,61,90,135,122)(16,57,21,101,62,91,136,123)(17,39,22,102,63,92,137,124)(18,40,23,103,64,93,138,125)(19,41,24,104,65,94,139,126), (20,135)(21,136)(22,137)(23,138)(24,139)(25,140)(26,141)(27,142)(28,143)(29,144)(30,145)(31,146)(32,147)(33,148)(34,149)(35,150)(36,151)(37,152)(38,134)(39,102)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,96)(53,97)(54,98)(55,99)(56,100)(57,101)(77,128)(78,129)(79,130)(80,131)(81,132)(82,133)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)>;
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), (1,42,25,105,66,95,140,127)(2,43,26,106,67,77,141,128)(3,44,27,107,68,78,142,129)(4,45,28,108,69,79,143,130)(5,46,29,109,70,80,144,131)(6,47,30,110,71,81,145,132)(7,48,31,111,72,82,146,133)(8,49,32,112,73,83,147,115)(9,50,33,113,74,84,148,116)(10,51,34,114,75,85,149,117)(11,52,35,96,76,86,150,118)(12,53,36,97,58,87,151,119)(13,54,37,98,59,88,152,120)(14,55,38,99,60,89,134,121)(15,56,20,100,61,90,135,122)(16,57,21,101,62,91,136,123)(17,39,22,102,63,92,137,124)(18,40,23,103,64,93,138,125)(19,41,24,104,65,94,139,126), (20,135)(21,136)(22,137)(23,138)(24,139)(25,140)(26,141)(27,142)(28,143)(29,144)(30,145)(31,146)(32,147)(33,148)(34,149)(35,150)(36,151)(37,152)(38,134)(39,102)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,113)(51,114)(52,96)(53,97)(54,98)(55,99)(56,100)(57,101)(77,128)(78,129)(79,130)(80,131)(81,132)(82,133)(83,115)(84,116)(85,117)(86,118)(87,119)(88,120)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127) );
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)], [(1,42,25,105,66,95,140,127),(2,43,26,106,67,77,141,128),(3,44,27,107,68,78,142,129),(4,45,28,108,69,79,143,130),(5,46,29,109,70,80,144,131),(6,47,30,110,71,81,145,132),(7,48,31,111,72,82,146,133),(8,49,32,112,73,83,147,115),(9,50,33,113,74,84,148,116),(10,51,34,114,75,85,149,117),(11,52,35,96,76,86,150,118),(12,53,36,97,58,87,151,119),(13,54,37,98,59,88,152,120),(14,55,38,99,60,89,134,121),(15,56,20,100,61,90,135,122),(16,57,21,101,62,91,136,123),(17,39,22,102,63,92,137,124),(18,40,23,103,64,93,138,125),(19,41,24,104,65,94,139,126)], [(20,135),(21,136),(22,137),(23,138),(24,139),(25,140),(26,141),(27,142),(28,143),(29,144),(30,145),(31,146),(32,147),(33,148),(34,149),(35,150),(36,151),(37,152),(38,134),(39,102),(40,103),(41,104),(42,105),(43,106),(44,107),(45,108),(46,109),(47,110),(48,111),(49,112),(50,113),(51,114),(52,96),(53,97),(54,98),(55,99),(56,100),(57,101),(77,128),(78,129),(79,130),(80,131),(81,132),(82,133),(83,115),(84,116),(85,117),(86,118),(87,119),(88,120),(89,121),(90,122),(91,123),(92,124),(93,125),(94,126),(95,127)]])
133 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 8A | 8B | 19A | ··· | 19R | 38A | ··· | 38R | 38S | ··· | 38AJ | 76A | ··· | 76R | 76S | ··· | 76AJ | 152A | ··· | 152AJ |
| order | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 19 | ··· | 19 | 38 | ··· | 38 | 38 | ··· | 38 | 76 | ··· | 76 | 76 | ··· | 76 | 152 | ··· | 152 |
| size | 1 | 1 | 4 | 2 | 4 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
133 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C19 | C38 | C38 | C38 | D4 | SD16 | D4×C19 | SD16×C19 |
| kernel | SD16×C19 | C152 | D4×C19 | Q8×C19 | SD16 | C8 | D4 | Q8 | C38 | C19 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 18 | 18 | 18 | 18 | 1 | 2 | 18 | 36 |
Matrix representation of SD16×C19 ►in GL2(𝔽457) generated by
| 256 | 0 |
| 0 | 256 |
| 210 | 247 |
| 210 | 210 |
| 1 | 0 |
| 0 | 456 |
G:=sub<GL(2,GF(457))| [256,0,0,256],[210,210,247,210],[1,0,0,456] >;
SD16×C19 in GAP, Magma, Sage, TeX
{\rm SD}_{16}\times C_{19} % in TeX
G:=Group("SD16xC19"); // GroupNames label
G:=SmallGroup(304,25);
// by ID
G=gap.SmallGroup(304,25);
# by ID
G:=PCGroup([5,-2,-2,-19,-2,-2,760,781,4563,2288,58]);
// 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^3>;
// generators/relations
Export