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G = D5×C38order 380 = 22·5·19

Direct product of C38 and D5

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

Aliases: D5×C38, C10⋊C38, C1903C2, C954C22, C5⋊(C2×C38), SmallGroup(380,9)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C38
C1C5C95D5×C19 — D5×C38
C5 — D5×C38
C1C38

Generators and relations for D5×C38
 G = < a,b,c | a38=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C2
5C22
5C38
5C38
5C2×C38

Smallest permutation representation of D5×C38
On 190 points
Generators in S190
(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)(153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190)
(1 95 131 172 45)(2 96 132 173 46)(3 97 133 174 47)(4 98 134 175 48)(5 99 135 176 49)(6 100 136 177 50)(7 101 137 178 51)(8 102 138 179 52)(9 103 139 180 53)(10 104 140 181 54)(11 105 141 182 55)(12 106 142 183 56)(13 107 143 184 57)(14 108 144 185 58)(15 109 145 186 59)(16 110 146 187 60)(17 111 147 188 61)(18 112 148 189 62)(19 113 149 190 63)(20 114 150 153 64)(21 77 151 154 65)(22 78 152 155 66)(23 79 115 156 67)(24 80 116 157 68)(25 81 117 158 69)(26 82 118 159 70)(27 83 119 160 71)(28 84 120 161 72)(29 85 121 162 73)(30 86 122 163 74)(31 87 123 164 75)(32 88 124 165 76)(33 89 125 166 39)(34 90 126 167 40)(35 91 127 168 41)(36 92 128 169 42)(37 93 129 170 43)(38 94 130 171 44)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 39)(15 40)(16 41)(17 42)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(29 54)(30 55)(31 56)(32 57)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(77 173)(78 174)(79 175)(80 176)(81 177)(82 178)(83 179)(84 180)(85 181)(86 182)(87 183)(88 184)(89 185)(90 186)(91 187)(92 188)(93 189)(94 190)(95 153)(96 154)(97 155)(98 156)(99 157)(100 158)(101 159)(102 160)(103 161)(104 162)(105 163)(106 164)(107 165)(108 166)(109 167)(110 168)(111 169)(112 170)(113 171)(114 172)(115 134)(116 135)(117 136)(118 137)(119 138)(120 139)(121 140)(122 141)(123 142)(124 143)(125 144)(126 145)(127 146)(128 147)(129 148)(130 149)(131 150)(132 151)(133 152)

G:=sub<Sym(190)| (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)(153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190), (1,95,131,172,45)(2,96,132,173,46)(3,97,133,174,47)(4,98,134,175,48)(5,99,135,176,49)(6,100,136,177,50)(7,101,137,178,51)(8,102,138,179,52)(9,103,139,180,53)(10,104,140,181,54)(11,105,141,182,55)(12,106,142,183,56)(13,107,143,184,57)(14,108,144,185,58)(15,109,145,186,59)(16,110,146,187,60)(17,111,147,188,61)(18,112,148,189,62)(19,113,149,190,63)(20,114,150,153,64)(21,77,151,154,65)(22,78,152,155,66)(23,79,115,156,67)(24,80,116,157,68)(25,81,117,158,69)(26,82,118,159,70)(27,83,119,160,71)(28,84,120,161,72)(29,85,121,162,73)(30,86,122,163,74)(31,87,123,164,75)(32,88,124,165,76)(33,89,125,166,39)(34,90,126,167,40)(35,91,127,168,41)(36,92,128,169,42)(37,93,129,170,43)(38,94,130,171,44), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(77,173)(78,174)(79,175)(80,176)(81,177)(82,178)(83,179)(84,180)(85,181)(86,182)(87,183)(88,184)(89,185)(90,186)(91,187)(92,188)(93,189)(94,190)(95,153)(96,154)(97,155)(98,156)(99,157)(100,158)(101,159)(102,160)(103,161)(104,162)(105,163)(106,164)(107,165)(108,166)(109,167)(110,168)(111,169)(112,170)(113,171)(114,172)(115,134)(116,135)(117,136)(118,137)(119,138)(120,139)(121,140)(122,141)(123,142)(124,143)(125,144)(126,145)(127,146)(128,147)(129,148)(130,149)(131,150)(132,151)(133,152)>;

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)(153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190), (1,95,131,172,45)(2,96,132,173,46)(3,97,133,174,47)(4,98,134,175,48)(5,99,135,176,49)(6,100,136,177,50)(7,101,137,178,51)(8,102,138,179,52)(9,103,139,180,53)(10,104,140,181,54)(11,105,141,182,55)(12,106,142,183,56)(13,107,143,184,57)(14,108,144,185,58)(15,109,145,186,59)(16,110,146,187,60)(17,111,147,188,61)(18,112,148,189,62)(19,113,149,190,63)(20,114,150,153,64)(21,77,151,154,65)(22,78,152,155,66)(23,79,115,156,67)(24,80,116,157,68)(25,81,117,158,69)(26,82,118,159,70)(27,83,119,160,71)(28,84,120,161,72)(29,85,121,162,73)(30,86,122,163,74)(31,87,123,164,75)(32,88,124,165,76)(33,89,125,166,39)(34,90,126,167,40)(35,91,127,168,41)(36,92,128,169,42)(37,93,129,170,43)(38,94,130,171,44), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,39)(15,40)(16,41)(17,42)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(29,54)(30,55)(31,56)(32,57)(33,58)(34,59)(35,60)(36,61)(37,62)(38,63)(77,173)(78,174)(79,175)(80,176)(81,177)(82,178)(83,179)(84,180)(85,181)(86,182)(87,183)(88,184)(89,185)(90,186)(91,187)(92,188)(93,189)(94,190)(95,153)(96,154)(97,155)(98,156)(99,157)(100,158)(101,159)(102,160)(103,161)(104,162)(105,163)(106,164)(107,165)(108,166)(109,167)(110,168)(111,169)(112,170)(113,171)(114,172)(115,134)(116,135)(117,136)(118,137)(119,138)(120,139)(121,140)(122,141)(123,142)(124,143)(125,144)(126,145)(127,146)(128,147)(129,148)(130,149)(131,150)(132,151)(133,152) );

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),(153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190)], [(1,95,131,172,45),(2,96,132,173,46),(3,97,133,174,47),(4,98,134,175,48),(5,99,135,176,49),(6,100,136,177,50),(7,101,137,178,51),(8,102,138,179,52),(9,103,139,180,53),(10,104,140,181,54),(11,105,141,182,55),(12,106,142,183,56),(13,107,143,184,57),(14,108,144,185,58),(15,109,145,186,59),(16,110,146,187,60),(17,111,147,188,61),(18,112,148,189,62),(19,113,149,190,63),(20,114,150,153,64),(21,77,151,154,65),(22,78,152,155,66),(23,79,115,156,67),(24,80,116,157,68),(25,81,117,158,69),(26,82,118,159,70),(27,83,119,160,71),(28,84,120,161,72),(29,85,121,162,73),(30,86,122,163,74),(31,87,123,164,75),(32,88,124,165,76),(33,89,125,166,39),(34,90,126,167,40),(35,91,127,168,41),(36,92,128,169,42),(37,93,129,170,43),(38,94,130,171,44)], [(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,72),(10,73),(11,74),(12,75),(13,76),(14,39),(15,40),(16,41),(17,42),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,49),(25,50),(26,51),(27,52),(28,53),(29,54),(30,55),(31,56),(32,57),(33,58),(34,59),(35,60),(36,61),(37,62),(38,63),(77,173),(78,174),(79,175),(80,176),(81,177),(82,178),(83,179),(84,180),(85,181),(86,182),(87,183),(88,184),(89,185),(90,186),(91,187),(92,188),(93,189),(94,190),(95,153),(96,154),(97,155),(98,156),(99,157),(100,158),(101,159),(102,160),(103,161),(104,162),(105,163),(106,164),(107,165),(108,166),(109,167),(110,168),(111,169),(112,170),(113,171),(114,172),(115,134),(116,135),(117,136),(118,137),(119,138),(120,139),(121,140),(122,141),(123,142),(124,143),(125,144),(126,145),(127,146),(128,147),(129,148),(130,149),(131,150),(132,151),(133,152)])

152 conjugacy classes

class 1 2A2B2C5A5B10A10B19A···19R38A···38R38S···38BB95A···95AJ190A···190AJ
order122255101019···1938···3838···3895···95190···190
size115522221···11···15···52···22···2

152 irreducible representations

dim1111112222
type+++++
imageC1C2C2C19C38C38D5D10D5×C19D5×C38
kernelD5×C38D5×C19C190D10D5C10C38C19C2C1
# reps121183618223636

Matrix representation of D5×C38 in GL2(𝔽191) generated by

1390
0139
,
01
19088
,
0190
1900
G:=sub<GL(2,GF(191))| [139,0,0,139],[0,190,1,88],[0,190,190,0] >;

D5×C38 in GAP, Magma, Sage, TeX

D_5\times C_{38}
% in TeX

G:=Group("D5xC38");
// GroupNames label

G:=SmallGroup(380,9);
// by ID

G=gap.SmallGroup(380,9);
# by ID

G:=PCGroup([4,-2,-2,-19,-5,4867]);
// Polycyclic

G:=Group<a,b,c|a^38=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×C38 in TeX

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