direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C37, C4⋊C74, C22⋊C74, C148⋊3C2, C74.6C22, (C2×C74)⋊1C2, C2.1(C2×C74), SmallGroup(296,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C37
G = < a,b,c | a37=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×C37
►On 148 points
Generators in S148
(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)
(1 141 55 76)(2 142 56 77)(3 143 57 78)(4 144 58 79)(5 145 59 80)(6 146 60 81)(7 147 61 82)(8 148 62 83)(9 112 63 84)(10 113 64 85)(11 114 65 86)(12 115 66 87)(13 116 67 88)(14 117 68 89)(15 118 69 90)(16 119 70 91)(17 120 71 92)(18 121 72 93)(19 122 73 94)(20 123 74 95)(21 124 38 96)(22 125 39 97)(23 126 40 98)(24 127 41 99)(25 128 42 100)(26 129 43 101)(27 130 44 102)(28 131 45 103)(29 132 46 104)(30 133 47 105)(31 134 48 106)(32 135 49 107)(33 136 50 108)(34 137 51 109)(35 138 52 110)(36 139 53 111)(37 140 54 75)
(75 140)(76 141)(77 142)(78 143)(79 144)(80 145)(81 146)(82 147)(83 148)(84 112)(85 113)(86 114)(87 115)(88 116)(89 117)(90 118)(91 119)(92 120)(93 121)(94 122)(95 123)(96 124)(97 125)(98 126)(99 127)(100 128)(101 129)(102 130)(103 131)(104 132)(105 133)(106 134)(107 135)(108 136)(109 137)(110 138)(111 139)
G:=sub<Sym(148)| (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), (1,141,55,76)(2,142,56,77)(3,143,57,78)(4,144,58,79)(5,145,59,80)(6,146,60,81)(7,147,61,82)(8,148,62,83)(9,112,63,84)(10,113,64,85)(11,114,65,86)(12,115,66,87)(13,116,67,88)(14,117,68,89)(15,118,69,90)(16,119,70,91)(17,120,71,92)(18,121,72,93)(19,122,73,94)(20,123,74,95)(21,124,38,96)(22,125,39,97)(23,126,40,98)(24,127,41,99)(25,128,42,100)(26,129,43,101)(27,130,44,102)(28,131,45,103)(29,132,46,104)(30,133,47,105)(31,134,48,106)(32,135,49,107)(33,136,50,108)(34,137,51,109)(35,138,52,110)(36,139,53,111)(37,140,54,75), (75,140)(76,141)(77,142)(78,143)(79,144)(80,145)(81,146)(82,147)(83,148)(84,112)(85,113)(86,114)(87,115)(88,116)(89,117)(90,118)(91,119)(92,120)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,127)(100,128)(101,129)(102,130)(103,131)(104,132)(105,133)(106,134)(107,135)(108,136)(109,137)(110,138)(111,139)>;
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), (1,141,55,76)(2,142,56,77)(3,143,57,78)(4,144,58,79)(5,145,59,80)(6,146,60,81)(7,147,61,82)(8,148,62,83)(9,112,63,84)(10,113,64,85)(11,114,65,86)(12,115,66,87)(13,116,67,88)(14,117,68,89)(15,118,69,90)(16,119,70,91)(17,120,71,92)(18,121,72,93)(19,122,73,94)(20,123,74,95)(21,124,38,96)(22,125,39,97)(23,126,40,98)(24,127,41,99)(25,128,42,100)(26,129,43,101)(27,130,44,102)(28,131,45,103)(29,132,46,104)(30,133,47,105)(31,134,48,106)(32,135,49,107)(33,136,50,108)(34,137,51,109)(35,138,52,110)(36,139,53,111)(37,140,54,75), (75,140)(76,141)(77,142)(78,143)(79,144)(80,145)(81,146)(82,147)(83,148)(84,112)(85,113)(86,114)(87,115)(88,116)(89,117)(90,118)(91,119)(92,120)(93,121)(94,122)(95,123)(96,124)(97,125)(98,126)(99,127)(100,128)(101,129)(102,130)(103,131)(104,132)(105,133)(106,134)(107,135)(108,136)(109,137)(110,138)(111,139) );
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)], [(1,141,55,76),(2,142,56,77),(3,143,57,78),(4,144,58,79),(5,145,59,80),(6,146,60,81),(7,147,61,82),(8,148,62,83),(9,112,63,84),(10,113,64,85),(11,114,65,86),(12,115,66,87),(13,116,67,88),(14,117,68,89),(15,118,69,90),(16,119,70,91),(17,120,71,92),(18,121,72,93),(19,122,73,94),(20,123,74,95),(21,124,38,96),(22,125,39,97),(23,126,40,98),(24,127,41,99),(25,128,42,100),(26,129,43,101),(27,130,44,102),(28,131,45,103),(29,132,46,104),(30,133,47,105),(31,134,48,106),(32,135,49,107),(33,136,50,108),(34,137,51,109),(35,138,52,110),(36,139,53,111),(37,140,54,75)], [(75,140),(76,141),(77,142),(78,143),(79,144),(80,145),(81,146),(82,147),(83,148),(84,112),(85,113),(86,114),(87,115),(88,116),(89,117),(90,118),(91,119),(92,120),(93,121),(94,122),(95,123),(96,124),(97,125),(98,126),(99,127),(100,128),(101,129),(102,130),(103,131),(104,132),(105,133),(106,134),(107,135),(108,136),(109,137),(110,138),(111,139)]])
185 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 37A | ··· | 37AJ | 74A | ··· | 74AJ | 74AK | ··· | 74DD | 148A | ··· | 148AJ |
| order | 1 | 2 | 2 | 2 | 4 | 37 | ··· | 37 | 74 | ··· | 74 | 74 | ··· | 74 | 148 | ··· | 148 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
185 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C37 | C74 | C74 | D4 | D4×C37 |
| kernel | D4×C37 | C148 | C2×C74 | D4 | C4 | C22 | C37 | C1 |
| # reps | 1 | 1 | 2 | 36 | 36 | 72 | 1 | 36 |
Matrix representation of D4×C37 ►in GL2(𝔽149) generated by
| 102 | 0 |
| 0 | 102 |
| 109 | 2 |
| 19 | 40 |
| 1 | 0 |
| 40 | 148 |
G:=sub<GL(2,GF(149))| [102,0,0,102],[109,19,2,40],[1,40,0,148] >;
D4×C37 in GAP, Magma, Sage, TeX
D_4\times C_{37} % in TeX
G:=Group("D4xC37"); // GroupNames label
G:=SmallGroup(296,10);
// by ID
G=gap.SmallGroup(296,10);
# by ID
G:=PCGroup([4,-2,-2,-37,-2,1201]);
// Polycyclic
G:=Group<a,b,c|a^37=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export