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G = C5×D37order 370 = 2·5·37

Direct product of C5 and D37

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

Aliases: C5×D37, C37⋊C10, C1852C2, SmallGroup(370,2)

Series: Derived Chief Lower central Upper central

C1C37 — C5×D37
C1C37C185 — C5×D37
C37 — C5×D37
C1C5

Generators and relations for C5×D37
 G = < a,b,c | a5=b37=c2=1, ab=ba, ac=ca, cbc=b-1 >

37C2
37C10

Smallest permutation representation of C5×D37
On 185 points
Generators in S185
(1 180 117 81 56)(2 181 118 82 57)(3 182 119 83 58)(4 183 120 84 59)(5 184 121 85 60)(6 185 122 86 61)(7 149 123 87 62)(8 150 124 88 63)(9 151 125 89 64)(10 152 126 90 65)(11 153 127 91 66)(12 154 128 92 67)(13 155 129 93 68)(14 156 130 94 69)(15 157 131 95 70)(16 158 132 96 71)(17 159 133 97 72)(18 160 134 98 73)(19 161 135 99 74)(20 162 136 100 38)(21 163 137 101 39)(22 164 138 102 40)(23 165 139 103 41)(24 166 140 104 42)(25 167 141 105 43)(26 168 142 106 44)(27 169 143 107 45)(28 170 144 108 46)(29 171 145 109 47)(30 172 146 110 48)(31 173 147 111 49)(32 174 148 75 50)(33 175 112 76 51)(34 176 113 77 52)(35 177 114 78 53)(36 178 115 79 54)(37 179 116 80 55)
(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 37)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)(38 73)(39 72)(40 71)(41 70)(42 69)(43 68)(44 67)(45 66)(46 65)(47 64)(48 63)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 56)(75 86)(76 85)(77 84)(78 83)(79 82)(80 81)(87 111)(88 110)(89 109)(90 108)(91 107)(92 106)(93 105)(94 104)(95 103)(96 102)(97 101)(98 100)(112 121)(113 120)(114 119)(115 118)(116 117)(122 148)(123 147)(124 146)(125 145)(126 144)(127 143)(128 142)(129 141)(130 140)(131 139)(132 138)(133 137)(134 136)(149 173)(150 172)(151 171)(152 170)(153 169)(154 168)(155 167)(156 166)(157 165)(158 164)(159 163)(160 162)(174 185)(175 184)(176 183)(177 182)(178 181)(179 180)

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

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

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

100 conjugacy classes

class 1  2 5A5B5C5D10A10B10C10D37A···37R185A···185BT
order1255551010101037···37185···185
size1371111373737372···22···2

100 irreducible representations

dim111122
type+++
imageC1C2C5C10D37C5×D37
kernelC5×D37C185D37C37C5C1
# reps11441872

Matrix representation of C5×D37 in GL2(𝔽1481) generated by

1360
0136
,
7191264
14801092
,
489914
568992
G:=sub<GL(2,GF(1481))| [136,0,0,136],[719,1480,1264,1092],[489,568,914,992] >;

C5×D37 in GAP, Magma, Sage, TeX

C_5\times D_{37}
% in TeX

G:=Group("C5xD37");
// GroupNames label

G:=SmallGroup(370,2);
// by ID

G=gap.SmallGroup(370,2);
# by ID

G:=PCGroup([3,-2,-5,-37,3242]);
// Polycyclic

G:=Group<a,b,c|a^5=b^37=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×D37 in TeX

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