direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D5×C37, C5⋊C74, C185⋊3C2, SmallGroup(370,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C37 |
Generators and relations for D5×C37
G = < a,b,c | a37=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C37
►On 185 points
Generators in S185
(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)(149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185)
(1 122 161 80 44)(2 123 162 81 45)(3 124 163 82 46)(4 125 164 83 47)(5 126 165 84 48)(6 127 166 85 49)(7 128 167 86 50)(8 129 168 87 51)(9 130 169 88 52)(10 131 170 89 53)(11 132 171 90 54)(12 133 172 91 55)(13 134 173 92 56)(14 135 174 93 57)(15 136 175 94 58)(16 137 176 95 59)(17 138 177 96 60)(18 139 178 97 61)(19 140 179 98 62)(20 141 180 99 63)(21 142 181 100 64)(22 143 182 101 65)(23 144 183 102 66)(24 145 184 103 67)(25 146 185 104 68)(26 147 149 105 69)(27 148 150 106 70)(28 112 151 107 71)(29 113 152 108 72)(30 114 153 109 73)(31 115 154 110 74)(32 116 155 111 38)(33 117 156 75 39)(34 118 157 76 40)(35 119 158 77 41)(36 120 159 78 42)(37 121 160 79 43)
(1 44)(2 45)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 59)(17 60)(18 61)(19 62)(20 63)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 71)(29 72)(30 73)(31 74)(32 38)(33 39)(34 40)(35 41)(36 42)(37 43)(75 117)(76 118)(77 119)(78 120)(79 121)(80 122)(81 123)(82 124)(83 125)(84 126)(85 127)(86 128)(87 129)(88 130)(89 131)(90 132)(91 133)(92 134)(93 135)(94 136)(95 137)(96 138)(97 139)(98 140)(99 141)(100 142)(101 143)(102 144)(103 145)(104 146)(105 147)(106 148)(107 112)(108 113)(109 114)(110 115)(111 116)
G:=sub<Sym(185)| (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)(149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185), (1,122,161,80,44)(2,123,162,81,45)(3,124,163,82,46)(4,125,164,83,47)(5,126,165,84,48)(6,127,166,85,49)(7,128,167,86,50)(8,129,168,87,51)(9,130,169,88,52)(10,131,170,89,53)(11,132,171,90,54)(12,133,172,91,55)(13,134,173,92,56)(14,135,174,93,57)(15,136,175,94,58)(16,137,176,95,59)(17,138,177,96,60)(18,139,178,97,61)(19,140,179,98,62)(20,141,180,99,63)(21,142,181,100,64)(22,143,182,101,65)(23,144,183,102,66)(24,145,184,103,67)(25,146,185,104,68)(26,147,149,105,69)(27,148,150,106,70)(28,112,151,107,71)(29,113,152,108,72)(30,114,153,109,73)(31,115,154,110,74)(32,116,155,111,38)(33,117,156,75,39)(34,118,157,76,40)(35,119,158,77,41)(36,120,159,78,42)(37,121,160,79,43), (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,38)(33,39)(34,40)(35,41)(36,42)(37,43)(75,117)(76,118)(77,119)(78,120)(79,121)(80,122)(81,123)(82,124)(83,125)(84,126)(85,127)(86,128)(87,129)(88,130)(89,131)(90,132)(91,133)(92,134)(93,135)(94,136)(95,137)(96,138)(97,139)(98,140)(99,141)(100,142)(101,143)(102,144)(103,145)(104,146)(105,147)(106,148)(107,112)(108,113)(109,114)(110,115)(111,116)>;
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)(149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185), (1,122,161,80,44)(2,123,162,81,45)(3,124,163,82,46)(4,125,164,83,47)(5,126,165,84,48)(6,127,166,85,49)(7,128,167,86,50)(8,129,168,87,51)(9,130,169,88,52)(10,131,170,89,53)(11,132,171,90,54)(12,133,172,91,55)(13,134,173,92,56)(14,135,174,93,57)(15,136,175,94,58)(16,137,176,95,59)(17,138,177,96,60)(18,139,178,97,61)(19,140,179,98,62)(20,141,180,99,63)(21,142,181,100,64)(22,143,182,101,65)(23,144,183,102,66)(24,145,184,103,67)(25,146,185,104,68)(26,147,149,105,69)(27,148,150,106,70)(28,112,151,107,71)(29,113,152,108,72)(30,114,153,109,73)(31,115,154,110,74)(32,116,155,111,38)(33,117,156,75,39)(34,118,157,76,40)(35,119,158,77,41)(36,120,159,78,42)(37,121,160,79,43), (1,44)(2,45)(3,46)(4,47)(5,48)(6,49)(7,50)(8,51)(9,52)(10,53)(11,54)(12,55)(13,56)(14,57)(15,58)(16,59)(17,60)(18,61)(19,62)(20,63)(21,64)(22,65)(23,66)(24,67)(25,68)(26,69)(27,70)(28,71)(29,72)(30,73)(31,74)(32,38)(33,39)(34,40)(35,41)(36,42)(37,43)(75,117)(76,118)(77,119)(78,120)(79,121)(80,122)(81,123)(82,124)(83,125)(84,126)(85,127)(86,128)(87,129)(88,130)(89,131)(90,132)(91,133)(92,134)(93,135)(94,136)(95,137)(96,138)(97,139)(98,140)(99,141)(100,142)(101,143)(102,144)(103,145)(104,146)(105,147)(106,148)(107,112)(108,113)(109,114)(110,115)(111,116) );
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),(149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185)], [(1,122,161,80,44),(2,123,162,81,45),(3,124,163,82,46),(4,125,164,83,47),(5,126,165,84,48),(6,127,166,85,49),(7,128,167,86,50),(8,129,168,87,51),(9,130,169,88,52),(10,131,170,89,53),(11,132,171,90,54),(12,133,172,91,55),(13,134,173,92,56),(14,135,174,93,57),(15,136,175,94,58),(16,137,176,95,59),(17,138,177,96,60),(18,139,178,97,61),(19,140,179,98,62),(20,141,180,99,63),(21,142,181,100,64),(22,143,182,101,65),(23,144,183,102,66),(24,145,184,103,67),(25,146,185,104,68),(26,147,149,105,69),(27,148,150,106,70),(28,112,151,107,71),(29,113,152,108,72),(30,114,153,109,73),(31,115,154,110,74),(32,116,155,111,38),(33,117,156,75,39),(34,118,157,76,40),(35,119,158,77,41),(36,120,159,78,42),(37,121,160,79,43)], [(1,44),(2,45),(3,46),(4,47),(5,48),(6,49),(7,50),(8,51),(9,52),(10,53),(11,54),(12,55),(13,56),(14,57),(15,58),(16,59),(17,60),(18,61),(19,62),(20,63),(21,64),(22,65),(23,66),(24,67),(25,68),(26,69),(27,70),(28,71),(29,72),(30,73),(31,74),(32,38),(33,39),(34,40),(35,41),(36,42),(37,43),(75,117),(76,118),(77,119),(78,120),(79,121),(80,122),(81,123),(82,124),(83,125),(84,126),(85,127),(86,128),(87,129),(88,130),(89,131),(90,132),(91,133),(92,134),(93,135),(94,136),(95,137),(96,138),(97,139),(98,140),(99,141),(100,142),(101,143),(102,144),(103,145),(104,146),(105,147),(106,148),(107,112),(108,113),(109,114),(110,115),(111,116)]])
148 conjugacy classes
| class | 1 | 2 | 5A | 5B | 37A | ··· | 37AJ | 74A | ··· | 74AJ | 185A | ··· | 185BT |
| order | 1 | 2 | 5 | 5 | 37 | ··· | 37 | 74 | ··· | 74 | 185 | ··· | 185 |
| size | 1 | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 |
148 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C37 | C74 | D5 | D5×C37 |
| kernel | D5×C37 | C185 | D5 | C5 | C37 | C1 |
| # reps | 1 | 1 | 36 | 36 | 2 | 72 |
Matrix representation of D5×C37 ►in GL2(𝔽1481) generated by
| 89 | 0 |
| 0 | 89 |
| 0 | 1 |
| 1480 | 38 |
| 1 | 0 |
| 38 | 1480 |
G:=sub<GL(2,GF(1481))| [89,0,0,89],[0,1480,1,38],[1,38,0,1480] >;
D5×C37 in GAP, Magma, Sage, TeX
D_5\times C_{37} % in TeX
G:=Group("D5xC37"); // GroupNames label
G:=SmallGroup(370,1);
// by ID
G=gap.SmallGroup(370,1);
# by ID
G:=PCGroup([3,-2,-37,-5,2666]);
// Polycyclic
G:=Group<a,b,c|a^37=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export