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G = C10×D19order 380 = 22·5·19

Direct product of C10 and D19

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

Aliases: C10×D19, C38⋊C10, C1902C2, C953C22, C19⋊(C2×C10), SmallGroup(380,8)

Series: Derived Chief Lower central Upper central

C1C19 — C10×D19
C1C19C95C5×D19 — C10×D19
C19 — C10×D19
C1C10

Generators and relations for C10×D19
 G = < a,b,c | a10=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >

19C2
19C2
19C22
19C10
19C10
19C2×C10

Smallest permutation representation of C10×D19
On 190 points
Generators in S190
(1 150 89 129 61 108 40 186 31 165)(2 151 90 130 62 109 41 187 32 166)(3 152 91 131 63 110 42 188 33 167)(4 134 92 132 64 111 43 189 34 168)(5 135 93 133 65 112 44 190 35 169)(6 136 94 115 66 113 45 172 36 170)(7 137 95 116 67 114 46 173 37 171)(8 138 77 117 68 96 47 174 38 153)(9 139 78 118 69 97 48 175 20 154)(10 140 79 119 70 98 49 176 21 155)(11 141 80 120 71 99 50 177 22 156)(12 142 81 121 72 100 51 178 23 157)(13 143 82 122 73 101 52 179 24 158)(14 144 83 123 74 102 53 180 25 159)(15 145 84 124 75 103 54 181 26 160)(16 146 85 125 76 104 55 182 27 161)(17 147 86 126 58 105 56 183 28 162)(18 148 87 127 59 106 57 184 29 163)(19 149 88 128 60 107 39 185 30 164)
(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)
(1 107)(2 106)(3 105)(4 104)(5 103)(6 102)(7 101)(8 100)(9 99)(10 98)(11 97)(12 96)(13 114)(14 113)(15 112)(16 111)(17 110)(18 109)(19 108)(20 120)(21 119)(22 118)(23 117)(24 116)(25 115)(26 133)(27 132)(28 131)(29 130)(30 129)(31 128)(32 127)(33 126)(34 125)(35 124)(36 123)(37 122)(38 121)(39 150)(40 149)(41 148)(42 147)(43 146)(44 145)(45 144)(46 143)(47 142)(48 141)(49 140)(50 139)(51 138)(52 137)(53 136)(54 135)(55 134)(56 152)(57 151)(58 167)(59 166)(60 165)(61 164)(62 163)(63 162)(64 161)(65 160)(66 159)(67 158)(68 157)(69 156)(70 155)(71 154)(72 153)(73 171)(74 170)(75 169)(76 168)(77 178)(78 177)(79 176)(80 175)(81 174)(82 173)(83 172)(84 190)(85 189)(86 188)(87 187)(88 186)(89 185)(90 184)(91 183)(92 182)(93 181)(94 180)(95 179)

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

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

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

110 conjugacy classes

class 1 2A2B2C5A5B5C5D10A10B10C10D10E···10L19A···19I38A···38I95A···95AJ190A···190AJ
order122255551010101010···1019···1938···3895···95190···190
size1119191111111119···192···22···22···22···2

110 irreducible representations

dim1111112222
type+++++
imageC1C2C2C5C10C10D19D38C5×D19C10×D19
kernelC10×D19C5×D19C190D38D19C38C10C5C2C1
# reps121484993636

Matrix representation of C10×D19 in GL2(𝔽191) generated by

1420
0142
,
77159
19055
,
114184
177
G:=sub<GL(2,GF(191))| [142,0,0,142],[77,190,159,55],[114,1,184,77] >;

C10×D19 in GAP, Magma, Sage, TeX

C_{10}\times D_{19}
% in TeX

G:=Group("C10xD19");
// GroupNames label

G:=SmallGroup(380,8);
// by ID

G=gap.SmallGroup(380,8);
# by ID

G:=PCGroup([4,-2,-2,-5,-19,5763]);
// Polycyclic

G:=Group<a,b,c|a^10=b^19=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C10×D19 in TeX

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