metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C37⋊2D4, C22⋊D37, D74⋊2C2, Dic37⋊C2, C2.5D74, C74.5C22, (C2×C74)⋊2C2, SmallGroup(296,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C37⋊D4
G = < a,b,c | a37=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C37⋊D4
►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 76 72 142)(2 75 73 141)(3 111 74 140)(4 110 38 139)(5 109 39 138)(6 108 40 137)(7 107 41 136)(8 106 42 135)(9 105 43 134)(10 104 44 133)(11 103 45 132)(12 102 46 131)(13 101 47 130)(14 100 48 129)(15 99 49 128)(16 98 50 127)(17 97 51 126)(18 96 52 125)(19 95 53 124)(20 94 54 123)(21 93 55 122)(22 92 56 121)(23 91 57 120)(24 90 58 119)(25 89 59 118)(26 88 60 117)(27 87 61 116)(28 86 62 115)(29 85 63 114)(30 84 64 113)(31 83 65 112)(32 82 66 148)(33 81 67 147)(34 80 68 146)(35 79 69 145)(36 78 70 144)(37 77 71 143)
(2 37)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 24)(16 23)(17 22)(18 21)(19 20)(38 69)(39 68)(40 67)(41 66)(42 65)(43 64)(44 63)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(70 74)(71 73)(75 143)(76 142)(77 141)(78 140)(79 139)(80 138)(81 137)(82 136)(83 135)(84 134)(85 133)(86 132)(87 131)(88 130)(89 129)(90 128)(91 127)(92 126)(93 125)(94 124)(95 123)(96 122)(97 121)(98 120)(99 119)(100 118)(101 117)(102 116)(103 115)(104 114)(105 113)(106 112)(107 148)(108 147)(109 146)(110 145)(111 144)
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), (1,76,72,142)(2,75,73,141)(3,111,74,140)(4,110,38,139)(5,109,39,138)(6,108,40,137)(7,107,41,136)(8,106,42,135)(9,105,43,134)(10,104,44,133)(11,103,45,132)(12,102,46,131)(13,101,47,130)(14,100,48,129)(15,99,49,128)(16,98,50,127)(17,97,51,126)(18,96,52,125)(19,95,53,124)(20,94,54,123)(21,93,55,122)(22,92,56,121)(23,91,57,120)(24,90,58,119)(25,89,59,118)(26,88,60,117)(27,87,61,116)(28,86,62,115)(29,85,63,114)(30,84,64,113)(31,83,65,112)(32,82,66,148)(33,81,67,147)(34,80,68,146)(35,79,69,145)(36,78,70,144)(37,77,71,143), (2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(70,74)(71,73)(75,143)(76,142)(77,141)(78,140)(79,139)(80,138)(81,137)(82,136)(83,135)(84,134)(85,133)(86,132)(87,131)(88,130)(89,129)(90,128)(91,127)(92,126)(93,125)(94,124)(95,123)(96,122)(97,121)(98,120)(99,119)(100,118)(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,148)(108,147)(109,146)(110,145)(111,144)>;
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), (1,76,72,142)(2,75,73,141)(3,111,74,140)(4,110,38,139)(5,109,39,138)(6,108,40,137)(7,107,41,136)(8,106,42,135)(9,105,43,134)(10,104,44,133)(11,103,45,132)(12,102,46,131)(13,101,47,130)(14,100,48,129)(15,99,49,128)(16,98,50,127)(17,97,51,126)(18,96,52,125)(19,95,53,124)(20,94,54,123)(21,93,55,122)(22,92,56,121)(23,91,57,120)(24,90,58,119)(25,89,59,118)(26,88,60,117)(27,87,61,116)(28,86,62,115)(29,85,63,114)(30,84,64,113)(31,83,65,112)(32,82,66,148)(33,81,67,147)(34,80,68,146)(35,79,69,145)(36,78,70,144)(37,77,71,143), (2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(70,74)(71,73)(75,143)(76,142)(77,141)(78,140)(79,139)(80,138)(81,137)(82,136)(83,135)(84,134)(85,133)(86,132)(87,131)(88,130)(89,129)(90,128)(91,127)(92,126)(93,125)(94,124)(95,123)(96,122)(97,121)(98,120)(99,119)(100,118)(101,117)(102,116)(103,115)(104,114)(105,113)(106,112)(107,148)(108,147)(109,146)(110,145)(111,144) );
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)], [(1,76,72,142),(2,75,73,141),(3,111,74,140),(4,110,38,139),(5,109,39,138),(6,108,40,137),(7,107,41,136),(8,106,42,135),(9,105,43,134),(10,104,44,133),(11,103,45,132),(12,102,46,131),(13,101,47,130),(14,100,48,129),(15,99,49,128),(16,98,50,127),(17,97,51,126),(18,96,52,125),(19,95,53,124),(20,94,54,123),(21,93,55,122),(22,92,56,121),(23,91,57,120),(24,90,58,119),(25,89,59,118),(26,88,60,117),(27,87,61,116),(28,86,62,115),(29,85,63,114),(30,84,64,113),(31,83,65,112),(32,82,66,148),(33,81,67,147),(34,80,68,146),(35,79,69,145),(36,78,70,144),(37,77,71,143)], [(2,37),(3,36),(4,35),(5,34),(6,33),(7,32),(8,31),(9,30),(10,29),(11,28),(12,27),(13,26),(14,25),(15,24),(16,23),(17,22),(18,21),(19,20),(38,69),(39,68),(40,67),(41,66),(42,65),(43,64),(44,63),(45,62),(46,61),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54),(70,74),(71,73),(75,143),(76,142),(77,141),(78,140),(79,139),(80,138),(81,137),(82,136),(83,135),(84,134),(85,133),(86,132),(87,131),(88,130),(89,129),(90,128),(91,127),(92,126),(93,125),(94,124),(95,123),(96,122),(97,121),(98,120),(99,119),(100,118),(101,117),(102,116),(103,115),(104,114),(105,113),(106,112),(107,148),(108,147),(109,146),(110,145),(111,144)]])
77 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 37A | ··· | 37R | 74A | ··· | 74BB |
| order | 1 | 2 | 2 | 2 | 4 | 37 | ··· | 37 | 74 | ··· | 74 |
| size | 1 | 1 | 2 | 74 | 74 | 2 | ··· | 2 | 2 | ··· | 2 |
77 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D37 | D74 | C37⋊D4 |
| kernel | C37⋊D4 | Dic37 | D74 | C2×C74 | C37 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 18 | 18 | 36 |
Matrix representation of C37⋊D4 ►in GL2(𝔽149) generated by
| 84 | 1 |
| 73 | 47 |
| 41 | 138 |
| 31 | 108 |
| 47 | 18 |
| 76 | 102 |
G:=sub<GL(2,GF(149))| [84,73,1,47],[41,31,138,108],[47,76,18,102] >;
C37⋊D4 in GAP, Magma, Sage, TeX
C_{37}\rtimes D_4 % in TeX
G:=Group("C37:D4"); // GroupNames label
G:=SmallGroup(296,8);
// by ID
G=gap.SmallGroup(296,8);
# by ID
G:=PCGroup([4,-2,-2,-2,-37,49,4611]);
// Polycyclic
G:=Group<a,b,c|a^37=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export