metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D38⋊C4, C2.2D76, C38.6D4, C22.6D38, (C2×C76)⋊1C2, (C2×C4)⋊1D19, C38.5(C2×C4), C2.5(C4×D19), C19⋊1(C22⋊C4), (C22×D19).C2, (C2×Dic19)⋊1C2, C2.2(C19⋊D4), (C2×C38).6C22, SmallGroup(304,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D38⋊C4
G = < a,b,c | a38=b2=c4=1, bab=a-1, ac=ca, cbc-1=a19b >
Smallest permutation representation of D38⋊C4
►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 44)(2 43)(3 42)(4 41)(5 40)(6 39)(7 76)(8 75)(9 74)(10 73)(11 72)(12 71)(13 70)(14 69)(15 68)(16 67)(17 66)(18 65)(19 64)(20 63)(21 62)(22 61)(23 60)(24 59)(25 58)(26 57)(27 56)(28 55)(29 54)(30 53)(31 52)(32 51)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(77 135)(78 134)(79 133)(80 132)(81 131)(82 130)(83 129)(84 128)(85 127)(86 126)(87 125)(88 124)(89 123)(90 122)(91 121)(92 120)(93 119)(94 118)(95 117)(96 116)(97 115)(98 152)(99 151)(100 150)(101 149)(102 148)(103 147)(104 146)(105 145)(106 144)(107 143)(108 142)(109 141)(110 140)(111 139)(112 138)(113 137)(114 136)
(1 119 45 113)(2 120 46 114)(3 121 47 77)(4 122 48 78)(5 123 49 79)(6 124 50 80)(7 125 51 81)(8 126 52 82)(9 127 53 83)(10 128 54 84)(11 129 55 85)(12 130 56 86)(13 131 57 87)(14 132 58 88)(15 133 59 89)(16 134 60 90)(17 135 61 91)(18 136 62 92)(19 137 63 93)(20 138 64 94)(21 139 65 95)(22 140 66 96)(23 141 67 97)(24 142 68 98)(25 143 69 99)(26 144 70 100)(27 145 71 101)(28 146 72 102)(29 147 73 103)(30 148 74 104)(31 149 75 105)(32 150 76 106)(33 151 39 107)(34 152 40 108)(35 115 41 109)(36 116 42 110)(37 117 43 111)(38 118 44 112)
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), (1,44)(2,43)(3,42)(4,41)(5,40)(6,39)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(77,135)(78,134)(79,133)(80,132)(81,131)(82,130)(83,129)(84,128)(85,127)(86,126)(87,125)(88,124)(89,123)(90,122)(91,121)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,152)(99,151)(100,150)(101,149)(102,148)(103,147)(104,146)(105,145)(106,144)(107,143)(108,142)(109,141)(110,140)(111,139)(112,138)(113,137)(114,136), (1,119,45,113)(2,120,46,114)(3,121,47,77)(4,122,48,78)(5,123,49,79)(6,124,50,80)(7,125,51,81)(8,126,52,82)(9,127,53,83)(10,128,54,84)(11,129,55,85)(12,130,56,86)(13,131,57,87)(14,132,58,88)(15,133,59,89)(16,134,60,90)(17,135,61,91)(18,136,62,92)(19,137,63,93)(20,138,64,94)(21,139,65,95)(22,140,66,96)(23,141,67,97)(24,142,68,98)(25,143,69,99)(26,144,70,100)(27,145,71,101)(28,146,72,102)(29,147,73,103)(30,148,74,104)(31,149,75,105)(32,150,76,106)(33,151,39,107)(34,152,40,108)(35,115,41,109)(36,116,42,110)(37,117,43,111)(38,118,44,112)>;
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), (1,44)(2,43)(3,42)(4,41)(5,40)(6,39)(7,76)(8,75)(9,74)(10,73)(11,72)(12,71)(13,70)(14,69)(15,68)(16,67)(17,66)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)(24,59)(25,58)(26,57)(27,56)(28,55)(29,54)(30,53)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(77,135)(78,134)(79,133)(80,132)(81,131)(82,130)(83,129)(84,128)(85,127)(86,126)(87,125)(88,124)(89,123)(90,122)(91,121)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,152)(99,151)(100,150)(101,149)(102,148)(103,147)(104,146)(105,145)(106,144)(107,143)(108,142)(109,141)(110,140)(111,139)(112,138)(113,137)(114,136), (1,119,45,113)(2,120,46,114)(3,121,47,77)(4,122,48,78)(5,123,49,79)(6,124,50,80)(7,125,51,81)(8,126,52,82)(9,127,53,83)(10,128,54,84)(11,129,55,85)(12,130,56,86)(13,131,57,87)(14,132,58,88)(15,133,59,89)(16,134,60,90)(17,135,61,91)(18,136,62,92)(19,137,63,93)(20,138,64,94)(21,139,65,95)(22,140,66,96)(23,141,67,97)(24,142,68,98)(25,143,69,99)(26,144,70,100)(27,145,71,101)(28,146,72,102)(29,147,73,103)(30,148,74,104)(31,149,75,105)(32,150,76,106)(33,151,39,107)(34,152,40,108)(35,115,41,109)(36,116,42,110)(37,117,43,111)(38,118,44,112) );
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)], [(1,44),(2,43),(3,42),(4,41),(5,40),(6,39),(7,76),(8,75),(9,74),(10,73),(11,72),(12,71),(13,70),(14,69),(15,68),(16,67),(17,66),(18,65),(19,64),(20,63),(21,62),(22,61),(23,60),(24,59),(25,58),(26,57),(27,56),(28,55),(29,54),(30,53),(31,52),(32,51),(33,50),(34,49),(35,48),(36,47),(37,46),(38,45),(77,135),(78,134),(79,133),(80,132),(81,131),(82,130),(83,129),(84,128),(85,127),(86,126),(87,125),(88,124),(89,123),(90,122),(91,121),(92,120),(93,119),(94,118),(95,117),(96,116),(97,115),(98,152),(99,151),(100,150),(101,149),(102,148),(103,147),(104,146),(105,145),(106,144),(107,143),(108,142),(109,141),(110,140),(111,139),(112,138),(113,137),(114,136)], [(1,119,45,113),(2,120,46,114),(3,121,47,77),(4,122,48,78),(5,123,49,79),(6,124,50,80),(7,125,51,81),(8,126,52,82),(9,127,53,83),(10,128,54,84),(11,129,55,85),(12,130,56,86),(13,131,57,87),(14,132,58,88),(15,133,59,89),(16,134,60,90),(17,135,61,91),(18,136,62,92),(19,137,63,93),(20,138,64,94),(21,139,65,95),(22,140,66,96),(23,141,67,97),(24,142,68,98),(25,143,69,99),(26,144,70,100),(27,145,71,101),(28,146,72,102),(29,147,73,103),(30,148,74,104),(31,149,75,105),(32,150,76,106),(33,151,39,107),(34,152,40,108),(35,115,41,109),(36,116,42,110),(37,117,43,111),(38,118,44,112)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 19A | ··· | 19I | 38A | ··· | 38AA | 76A | ··· | 76AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 1 | 1 | 38 | 38 | 2 | 2 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D19 | D38 | C4×D19 | D76 | C19⋊D4 |
| kernel | D38⋊C4 | C2×Dic19 | C2×C76 | C22×D19 | D38 | C38 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 2 | 9 | 9 | 18 | 18 | 18 |
Matrix representation of D38⋊C4 ►in GL3(𝔽229) generated by
| 1 | 0 | 0 |
| 0 | 32 | 32 |
| 0 | 197 | 104 |
| 1 | 0 | 0 |
| 0 | 197 | 197 |
| 0 | 125 | 32 |
| 122 | 0 | 0 |
| 0 | 164 | 185 |
| 0 | 44 | 65 |
G:=sub<GL(3,GF(229))| [1,0,0,0,32,197,0,32,104],[1,0,0,0,197,125,0,197,32],[122,0,0,0,164,44,0,185,65] >;
D38⋊C4 in GAP, Magma, Sage, TeX
D_{38}\rtimes C_4 % in TeX
G:=Group("D38:C4"); // GroupNames label
G:=SmallGroup(304,13);
// by ID
G=gap.SmallGroup(304,13);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,101,26,7204]);
// Polycyclic
G:=Group<a,b,c|a^38=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^19*b>;
// generators/relations
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