direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C38, C23⋊C38, C76⋊4C22, C38.11C23, C4⋊(C2×C38), (C2×C4)⋊2C38, (C2×C76)⋊6C2, C22⋊(C2×C38), (C22×C38)⋊1C2, (C2×C38)⋊2C22, C2.1(C22×C38), SmallGroup(304,38)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C38
G = < a,b,c | a38=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 70 in 54 conjugacy classes, 38 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, D4, C23, C2×D4, C19, C38, C38, C38, C76, C2×C38, C2×C38, C2×C38, C2×C76, D4×C19, C22×C38, D4×C38
Quotients: C1, C2, C22, D4, C23, C2×D4, C19, C38, C2×C38, D4×C19, C22×C38, D4×C38
►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 73 112 123)(2 74 113 124)(3 75 114 125)(4 76 77 126)(5 39 78 127)(6 40 79 128)(7 41 80 129)(8 42 81 130)(9 43 82 131)(10 44 83 132)(11 45 84 133)(12 46 85 134)(13 47 86 135)(14 48 87 136)(15 49 88 137)(16 50 89 138)(17 51 90 139)(18 52 91 140)(19 53 92 141)(20 54 93 142)(21 55 94 143)(22 56 95 144)(23 57 96 145)(24 58 97 146)(25 59 98 147)(26 60 99 148)(27 61 100 149)(28 62 101 150)(29 63 102 151)(30 64 103 152)(31 65 104 115)(32 66 105 116)(33 67 106 117)(34 68 107 118)(35 69 108 119)(36 70 109 120)(37 71 110 121)(38 72 111 122)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 61)(9 62)(10 63)(11 64)(12 65)(13 66)(14 67)(15 68)(16 69)(17 70)(18 71)(19 72)(20 73)(21 74)(22 75)(23 76)(24 39)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(77 145)(78 146)(79 147)(80 148)(81 149)(82 150)(83 151)(84 152)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)(91 121)(92 122)(93 123)(94 124)(95 125)(96 126)(97 127)(98 128)(99 129)(100 130)(101 131)(102 132)(103 133)(104 134)(105 135)(106 136)(107 137)(108 138)(109 139)(110 140)(111 141)(112 142)(113 143)(114 144)
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), (1,73,112,123)(2,74,113,124)(3,75,114,125)(4,76,77,126)(5,39,78,127)(6,40,79,128)(7,41,80,129)(8,42,81,130)(9,43,82,131)(10,44,83,132)(11,45,84,133)(12,46,85,134)(13,47,86,135)(14,48,87,136)(15,49,88,137)(16,50,89,138)(17,51,90,139)(18,52,91,140)(19,53,92,141)(20,54,93,142)(21,55,94,143)(22,56,95,144)(23,57,96,145)(24,58,97,146)(25,59,98,147)(26,60,99,148)(27,61,100,149)(28,62,101,150)(29,63,102,151)(30,64,103,152)(31,65,104,115)(32,66,105,116)(33,67,106,117)(34,68,107,118)(35,69,108,119)(36,70,109,120)(37,71,110,121)(38,72,111,122), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,39)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(77,145)(78,146)(79,147)(80,148)(81,149)(82,150)(83,151)(84,152)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(97,127)(98,128)(99,129)(100,130)(101,131)(102,132)(103,133)(104,134)(105,135)(106,136)(107,137)(108,138)(109,139)(110,140)(111,141)(112,142)(113,143)(114,144)>;
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), (1,73,112,123)(2,74,113,124)(3,75,114,125)(4,76,77,126)(5,39,78,127)(6,40,79,128)(7,41,80,129)(8,42,81,130)(9,43,82,131)(10,44,83,132)(11,45,84,133)(12,46,85,134)(13,47,86,135)(14,48,87,136)(15,49,88,137)(16,50,89,138)(17,51,90,139)(18,52,91,140)(19,53,92,141)(20,54,93,142)(21,55,94,143)(22,56,95,144)(23,57,96,145)(24,58,97,146)(25,59,98,147)(26,60,99,148)(27,61,100,149)(28,62,101,150)(29,63,102,151)(30,64,103,152)(31,65,104,115)(32,66,105,116)(33,67,106,117)(34,68,107,118)(35,69,108,119)(36,70,109,120)(37,71,110,121)(38,72,111,122), (1,54)(2,55)(3,56)(4,57)(5,58)(6,59)(7,60)(8,61)(9,62)(10,63)(11,64)(12,65)(13,66)(14,67)(15,68)(16,69)(17,70)(18,71)(19,72)(20,73)(21,74)(22,75)(23,76)(24,39)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,49)(35,50)(36,51)(37,52)(38,53)(77,145)(78,146)(79,147)(80,148)(81,149)(82,150)(83,151)(84,152)(85,115)(86,116)(87,117)(88,118)(89,119)(90,120)(91,121)(92,122)(93,123)(94,124)(95,125)(96,126)(97,127)(98,128)(99,129)(100,130)(101,131)(102,132)(103,133)(104,134)(105,135)(106,136)(107,137)(108,138)(109,139)(110,140)(111,141)(112,142)(113,143)(114,144) );
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)], [(1,73,112,123),(2,74,113,124),(3,75,114,125),(4,76,77,126),(5,39,78,127),(6,40,79,128),(7,41,80,129),(8,42,81,130),(9,43,82,131),(10,44,83,132),(11,45,84,133),(12,46,85,134),(13,47,86,135),(14,48,87,136),(15,49,88,137),(16,50,89,138),(17,51,90,139),(18,52,91,140),(19,53,92,141),(20,54,93,142),(21,55,94,143),(22,56,95,144),(23,57,96,145),(24,58,97,146),(25,59,98,147),(26,60,99,148),(27,61,100,149),(28,62,101,150),(29,63,102,151),(30,64,103,152),(31,65,104,115),(32,66,105,116),(33,67,106,117),(34,68,107,118),(35,69,108,119),(36,70,109,120),(37,71,110,121),(38,72,111,122)], [(1,54),(2,55),(3,56),(4,57),(5,58),(6,59),(7,60),(8,61),(9,62),(10,63),(11,64),(12,65),(13,66),(14,67),(15,68),(16,69),(17,70),(18,71),(19,72),(20,73),(21,74),(22,75),(23,76),(24,39),(25,40),(26,41),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(33,48),(34,49),(35,50),(36,51),(37,52),(38,53),(77,145),(78,146),(79,147),(80,148),(81,149),(82,150),(83,151),(84,152),(85,115),(86,116),(87,117),(88,118),(89,119),(90,120),(91,121),(92,122),(93,123),(94,124),(95,125),(96,126),(97,127),(98,128),(99,129),(100,130),(101,131),(102,132),(103,133),(104,134),(105,135),(106,136),(107,137),(108,138),(109,139),(110,140),(111,141),(112,142),(113,143),(114,144)]])
190 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 19A | ··· | 19R | 38A | ··· | 38BB | 38BC | ··· | 38DV | 76A | ··· | 76AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 19 | ··· | 19 | 38 | ··· | 38 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
190 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C19 | C38 | C38 | C38 | D4 | D4×C19 |
| kernel | D4×C38 | C2×C76 | D4×C19 | C22×C38 | C2×D4 | C2×C4 | D4 | C23 | C38 | C2 |
| # reps | 1 | 1 | 4 | 2 | 18 | 18 | 72 | 36 | 2 | 36 |
Matrix representation of D4×C38 ►in GL3(𝔽229) generated by
| 228 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 11 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 228 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(229))| [228,0,0,0,11,0,0,0,11],[1,0,0,0,0,228,0,1,0],[1,0,0,0,0,1,0,1,0] >;
D4×C38 in GAP, Magma, Sage, TeX
D_4\times C_{38} % in TeX
G:=Group("D4xC38"); // GroupNames label
G:=SmallGroup(304,38);
// by ID
G=gap.SmallGroup(304,38);
# by ID
G:=PCGroup([5,-2,-2,-2,-19,-2,1541]);
// Polycyclic
G:=Group<a,b,c|a^38=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations