direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic19, C38⋊C4, C2.2D38, C22.D19, C38.4C22, C19⋊2(C2×C4), (C2×C38).C2, SmallGroup(152,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C2×Dic19 |
Generators and relations for C2×Dic19
G = < a,b,c | a2=b38=1, c2=b19, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic19
►Regular action on 152 points
Generators in S152
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)(37 57)(38 58)(77 134)(78 135)(79 136)(80 137)(81 138)(82 139)(83 140)(84 141)(85 142)(86 143)(87 144)(88 145)(89 146)(90 147)(91 148)(92 149)(93 150)(94 151)(95 152)(96 115)(97 116)(98 117)(99 118)(100 119)(101 120)(102 121)(103 122)(104 123)(105 124)(106 125)(107 126)(108 127)(109 128)(110 129)(111 130)(112 131)(113 132)(114 133)
(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 96 20 77)(2 95 21 114)(3 94 22 113)(4 93 23 112)(5 92 24 111)(6 91 25 110)(7 90 26 109)(8 89 27 108)(9 88 28 107)(10 87 29 106)(11 86 30 105)(12 85 31 104)(13 84 32 103)(14 83 33 102)(15 82 34 101)(16 81 35 100)(17 80 36 99)(18 79 37 98)(19 78 38 97)(39 135 58 116)(40 134 59 115)(41 133 60 152)(42 132 61 151)(43 131 62 150)(44 130 63 149)(45 129 64 148)(46 128 65 147)(47 127 66 146)(48 126 67 145)(49 125 68 144)(50 124 69 143)(51 123 70 142)(52 122 71 141)(53 121 72 140)(54 120 73 139)(55 119 74 138)(56 118 75 137)(57 117 76 136)
G:=sub<Sym(152)| (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(77,134)(78,135)(79,136)(80,137)(81,138)(82,139)(83,140)(84,141)(85,142)(86,143)(87,144)(88,145)(89,146)(90,147)(91,148)(92,149)(93,150)(94,151)(95,152)(96,115)(97,116)(98,117)(99,118)(100,119)(101,120)(102,121)(103,122)(104,123)(105,124)(106,125)(107,126)(108,127)(109,128)(110,129)(111,130)(112,131)(113,132)(114,133), (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), (1,96,20,77)(2,95,21,114)(3,94,22,113)(4,93,23,112)(5,92,24,111)(6,91,25,110)(7,90,26,109)(8,89,27,108)(9,88,28,107)(10,87,29,106)(11,86,30,105)(12,85,31,104)(13,84,32,103)(14,83,33,102)(15,82,34,101)(16,81,35,100)(17,80,36,99)(18,79,37,98)(19,78,38,97)(39,135,58,116)(40,134,59,115)(41,133,60,152)(42,132,61,151)(43,131,62,150)(44,130,63,149)(45,129,64,148)(46,128,65,147)(47,127,66,146)(48,126,67,145)(49,125,68,144)(50,124,69,143)(51,123,70,142)(52,122,71,141)(53,121,72,140)(54,120,73,139)(55,119,74,138)(56,118,75,137)(57,117,76,136)>;
G:=Group( (1,59)(2,60)(3,61)(4,62)(5,63)(6,64)(7,65)(8,66)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,73)(16,74)(17,75)(18,76)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,57)(38,58)(77,134)(78,135)(79,136)(80,137)(81,138)(82,139)(83,140)(84,141)(85,142)(86,143)(87,144)(88,145)(89,146)(90,147)(91,148)(92,149)(93,150)(94,151)(95,152)(96,115)(97,116)(98,117)(99,118)(100,119)(101,120)(102,121)(103,122)(104,123)(105,124)(106,125)(107,126)(108,127)(109,128)(110,129)(111,130)(112,131)(113,132)(114,133), (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), (1,96,20,77)(2,95,21,114)(3,94,22,113)(4,93,23,112)(5,92,24,111)(6,91,25,110)(7,90,26,109)(8,89,27,108)(9,88,28,107)(10,87,29,106)(11,86,30,105)(12,85,31,104)(13,84,32,103)(14,83,33,102)(15,82,34,101)(16,81,35,100)(17,80,36,99)(18,79,37,98)(19,78,38,97)(39,135,58,116)(40,134,59,115)(41,133,60,152)(42,132,61,151)(43,131,62,150)(44,130,63,149)(45,129,64,148)(46,128,65,147)(47,127,66,146)(48,126,67,145)(49,125,68,144)(50,124,69,143)(51,123,70,142)(52,122,71,141)(53,121,72,140)(54,120,73,139)(55,119,74,138)(56,118,75,137)(57,117,76,136) );
G=PermutationGroup([[(1,59),(2,60),(3,61),(4,62),(5,63),(6,64),(7,65),(8,66),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,73),(16,74),(17,75),(18,76),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,55),(36,56),(37,57),(38,58),(77,134),(78,135),(79,136),(80,137),(81,138),(82,139),(83,140),(84,141),(85,142),(86,143),(87,144),(88,145),(89,146),(90,147),(91,148),(92,149),(93,150),(94,151),(95,152),(96,115),(97,116),(98,117),(99,118),(100,119),(101,120),(102,121),(103,122),(104,123),(105,124),(106,125),(107,126),(108,127),(109,128),(110,129),(111,130),(112,131),(113,132),(114,133)], [(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)], [(1,96,20,77),(2,95,21,114),(3,94,22,113),(4,93,23,112),(5,92,24,111),(6,91,25,110),(7,90,26,109),(8,89,27,108),(9,88,28,107),(10,87,29,106),(11,86,30,105),(12,85,31,104),(13,84,32,103),(14,83,33,102),(15,82,34,101),(16,81,35,100),(17,80,36,99),(18,79,37,98),(19,78,38,97),(39,135,58,116),(40,134,59,115),(41,133,60,152),(42,132,61,151),(43,131,62,150),(44,130,63,149),(45,129,64,148),(46,128,65,147),(47,127,66,146),(48,126,67,145),(49,125,68,144),(50,124,69,143),(51,123,70,142),(52,122,71,141),(53,121,72,140),(54,120,73,139),(55,119,74,138),(56,118,75,137),(57,117,76,136)]])
C2×Dic19 is a maximal subgroup of
Dic19⋊C4 C76⋊C4 D38⋊C4 C23.D19 C2×C4×D19 D4⋊2D19
C2×Dic19 is a maximal quotient of C76.C4 C76⋊C4 C23.D19
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 19A | ··· | 19I | 38A | ··· | 38AA |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 19 | ··· | 19 | 38 | ··· | 38 |
| size | 1 | 1 | 1 | 1 | 19 | 19 | 19 | 19 | 2 | ··· | 2 | 2 | ··· | 2 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | D19 | Dic19 | D38 |
| kernel | C2×Dic19 | Dic19 | C2×C38 | C38 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 9 | 18 | 9 |
Matrix representation of C2×Dic19 ►in GL3(𝔽229) generated by
| 228 | 0 | 0 |
| 0 | 228 | 0 |
| 0 | 0 | 228 |
| 1 | 0 | 0 |
| 0 | 0 | 228 |
| 0 | 1 | 55 |
| 228 | 0 | 0 |
| 0 | 57 | 31 |
| 0 | 102 | 172 |
G:=sub<GL(3,GF(229))| [228,0,0,0,228,0,0,0,228],[1,0,0,0,0,1,0,228,55],[228,0,0,0,57,102,0,31,172] >;
C2×Dic19 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{19} % in TeX
G:=Group("C2xDic19"); // GroupNames label
G:=SmallGroup(152,6);
// by ID
G=gap.SmallGroup(152,6);
# by ID
G:=PCGroup([4,-2,-2,-2,-19,16,2307]);
// Polycyclic
G:=Group<a,b,c|a^2=b^38=1,c^2=b^19,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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