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G = D5×C37order 370 = 2·5·37

Direct product of C37 and D5

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: D5×C37, C5⋊C74, C1853C2, SmallGroup(370,1)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C37
C1C5C185 — D5×C37
C5 — D5×C37
C1C37

Generators and relations for D5×C37
 G = < a,b,c | a37=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C74

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 76 171 135 55)(2 77 172 136 56)(3 78 173 137 57)(4 79 174 138 58)(5 80 175 139 59)(6 81 176 140 60)(7 82 177 141 61)(8 83 178 142 62)(9 84 179 143 63)(10 85 180 144 64)(11 86 181 145 65)(12 87 182 146 66)(13 88 183 147 67)(14 89 184 148 68)(15 90 185 112 69)(16 91 149 113 70)(17 92 150 114 71)(18 93 151 115 72)(19 94 152 116 73)(20 95 153 117 74)(21 96 154 118 38)(22 97 155 119 39)(23 98 156 120 40)(24 99 157 121 41)(25 100 158 122 42)(26 101 159 123 43)(27 102 160 124 44)(28 103 161 125 45)(29 104 162 126 46)(30 105 163 127 47)(31 106 164 128 48)(32 107 165 129 49)(33 108 166 130 50)(34 109 167 131 51)(35 110 168 132 52)(36 111 169 133 53)(37 75 170 134 54)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 73)(20 74)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 53)(37 54)(75 134)(76 135)(77 136)(78 137)(79 138)(80 139)(81 140)(82 141)(83 142)(84 143)(85 144)(86 145)(87 146)(88 147)(89 148)(90 112)(91 113)(92 114)(93 115)(94 116)(95 117)(96 118)(97 119)(98 120)(99 121)(100 122)(101 123)(102 124)(103 125)(104 126)(105 127)(106 128)(107 129)(108 130)(109 131)(110 132)(111 133)

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), (1,76,171,135,55)(2,77,172,136,56)(3,78,173,137,57)(4,79,174,138,58)(5,80,175,139,59)(6,81,176,140,60)(7,82,177,141,61)(8,83,178,142,62)(9,84,179,143,63)(10,85,180,144,64)(11,86,181,145,65)(12,87,182,146,66)(13,88,183,147,67)(14,89,184,148,68)(15,90,185,112,69)(16,91,149,113,70)(17,92,150,114,71)(18,93,151,115,72)(19,94,152,116,73)(20,95,153,117,74)(21,96,154,118,38)(22,97,155,119,39)(23,98,156,120,40)(24,99,157,121,41)(25,100,158,122,42)(26,101,159,123,43)(27,102,160,124,44)(28,103,161,125,45)(29,104,162,126,46)(30,105,163,127,47)(31,106,164,128,48)(32,107,165,129,49)(33,108,166,130,50)(34,109,167,131,51)(35,110,168,132,52)(36,111,169,133,53)(37,75,170,134,54), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,53)(37,54)(75,134)(76,135)(77,136)(78,137)(79,138)(80,139)(81,140)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,112)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(111,133)>;

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), (1,76,171,135,55)(2,77,172,136,56)(3,78,173,137,57)(4,79,174,138,58)(5,80,175,139,59)(6,81,176,140,60)(7,82,177,141,61)(8,83,178,142,62)(9,84,179,143,63)(10,85,180,144,64)(11,86,181,145,65)(12,87,182,146,66)(13,88,183,147,67)(14,89,184,148,68)(15,90,185,112,69)(16,91,149,113,70)(17,92,150,114,71)(18,93,151,115,72)(19,94,152,116,73)(20,95,153,117,74)(21,96,154,118,38)(22,97,155,119,39)(23,98,156,120,40)(24,99,157,121,41)(25,100,158,122,42)(26,101,159,123,43)(27,102,160,124,44)(28,103,161,125,45)(29,104,162,126,46)(30,105,163,127,47)(31,106,164,128,48)(32,107,165,129,49)(33,108,166,130,50)(34,109,167,131,51)(35,110,168,132,52)(36,111,169,133,53)(37,75,170,134,54), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,73)(20,74)(21,38)(22,39)(23,40)(24,41)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,53)(37,54)(75,134)(76,135)(77,136)(78,137)(79,138)(80,139)(81,140)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,112)(91,113)(92,114)(93,115)(94,116)(95,117)(96,118)(97,119)(98,120)(99,121)(100,122)(101,123)(102,124)(103,125)(104,126)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(111,133) );

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)], [(1,76,171,135,55),(2,77,172,136,56),(3,78,173,137,57),(4,79,174,138,58),(5,80,175,139,59),(6,81,176,140,60),(7,82,177,141,61),(8,83,178,142,62),(9,84,179,143,63),(10,85,180,144,64),(11,86,181,145,65),(12,87,182,146,66),(13,88,183,147,67),(14,89,184,148,68),(15,90,185,112,69),(16,91,149,113,70),(17,92,150,114,71),(18,93,151,115,72),(19,94,152,116,73),(20,95,153,117,74),(21,96,154,118,38),(22,97,155,119,39),(23,98,156,120,40),(24,99,157,121,41),(25,100,158,122,42),(26,101,159,123,43),(27,102,160,124,44),(28,103,161,125,45),(29,104,162,126,46),(30,105,163,127,47),(31,106,164,128,48),(32,107,165,129,49),(33,108,166,130,50),(34,109,167,131,51),(35,110,168,132,52),(36,111,169,133,53),(37,75,170,134,54)], [(1,55),(2,56),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,63),(10,64),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,73),(20,74),(21,38),(22,39),(23,40),(24,41),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,49),(33,50),(34,51),(35,52),(36,53),(37,54),(75,134),(76,135),(77,136),(78,137),(79,138),(80,139),(81,140),(82,141),(83,142),(84,143),(85,144),(86,145),(87,146),(88,147),(89,148),(90,112),(91,113),(92,114),(93,115),(94,116),(95,117),(96,118),(97,119),(98,120),(99,121),(100,122),(101,123),(102,124),(103,125),(104,126),(105,127),(106,128),(107,129),(108,130),(109,131),(110,132),(111,133)])

148 conjugacy classes

class 1  2 5A5B37A···37AJ74A···74AJ185A···185BT
order125537···3774···74185···185
size15221···15···52···2

148 irreducible representations

dim111122
type+++
imageC1C2C37C74D5D5×C37
kernelD5×C37C185D5C5C37C1
# reps113636272

Matrix representation of D5×C37 in GL2(𝔽1481) generated by

890
089
,
01
148038
,
10
381480
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

Subgroup lattice of D5×C37 in TeX

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