direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C10×D19, C38⋊C10, C190⋊2C2, C95⋊3C22, C19⋊(C2×C10), SmallGroup(380,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C10×D19 |
Generators and relations for C10×D19
G = < a,b,c | a10=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C10×D19
►On 190 points
Generators in S190
(1 137 93 115 61 96 48 187 21 170)(2 138 94 116 62 97 49 188 22 171)(3 139 95 117 63 98 50 189 23 153)(4 140 77 118 64 99 51 190 24 154)(5 141 78 119 65 100 52 172 25 155)(6 142 79 120 66 101 53 173 26 156)(7 143 80 121 67 102 54 174 27 157)(8 144 81 122 68 103 55 175 28 158)(9 145 82 123 69 104 56 176 29 159)(10 146 83 124 70 105 57 177 30 160)(11 147 84 125 71 106 39 178 31 161)(12 148 85 126 72 107 40 179 32 162)(13 149 86 127 73 108 41 180 33 163)(14 150 87 128 74 109 42 181 34 164)(15 151 88 129 75 110 43 182 35 165)(16 152 89 130 76 111 44 183 36 166)(17 134 90 131 58 112 45 184 37 167)(18 135 91 132 59 113 46 185 38 168)(19 136 92 133 60 114 47 186 20 169)
(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)
(1 114)(2 113)(3 112)(4 111)(5 110)(6 109)(7 108)(8 107)(9 106)(10 105)(11 104)(12 103)(13 102)(14 101)(15 100)(16 99)(17 98)(18 97)(19 96)(20 115)(21 133)(22 132)(23 131)(24 130)(25 129)(26 128)(27 127)(28 126)(29 125)(30 124)(31 123)(32 122)(33 121)(34 120)(35 119)(36 118)(37 117)(38 116)(39 145)(40 144)(41 143)(42 142)(43 141)(44 140)(45 139)(46 138)(47 137)(48 136)(49 135)(50 134)(51 152)(52 151)(53 150)(54 149)(55 148)(56 147)(57 146)(58 153)(59 171)(60 170)(61 169)(62 168)(63 167)(64 166)(65 165)(66 164)(67 163)(68 162)(69 161)(70 160)(71 159)(72 158)(73 157)(74 156)(75 155)(76 154)(77 183)(78 182)(79 181)(80 180)(81 179)(82 178)(83 177)(84 176)(85 175)(86 174)(87 173)(88 172)(89 190)(90 189)(91 188)(92 187)(93 186)(94 185)(95 184)
G:=sub<Sym(190)| (1,137,93,115,61,96,48,187,21,170)(2,138,94,116,62,97,49,188,22,171)(3,139,95,117,63,98,50,189,23,153)(4,140,77,118,64,99,51,190,24,154)(5,141,78,119,65,100,52,172,25,155)(6,142,79,120,66,101,53,173,26,156)(7,143,80,121,67,102,54,174,27,157)(8,144,81,122,68,103,55,175,28,158)(9,145,82,123,69,104,56,176,29,159)(10,146,83,124,70,105,57,177,30,160)(11,147,84,125,71,106,39,178,31,161)(12,148,85,126,72,107,40,179,32,162)(13,149,86,127,73,108,41,180,33,163)(14,150,87,128,74,109,42,181,34,164)(15,151,88,129,75,110,43,182,35,165)(16,152,89,130,76,111,44,183,36,166)(17,134,90,131,58,112,45,184,37,167)(18,135,91,132,59,113,46,185,38,168)(19,136,92,133,60,114,47,186,20,169), (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), (1,114)(2,113)(3,112)(4,111)(5,110)(6,109)(7,108)(8,107)(9,106)(10,105)(11,104)(12,103)(13,102)(14,101)(15,100)(16,99)(17,98)(18,97)(19,96)(20,115)(21,133)(22,132)(23,131)(24,130)(25,129)(26,128)(27,127)(28,126)(29,125)(30,124)(31,123)(32,122)(33,121)(34,120)(35,119)(36,118)(37,117)(38,116)(39,145)(40,144)(41,143)(42,142)(43,141)(44,140)(45,139)(46,138)(47,137)(48,136)(49,135)(50,134)(51,152)(52,151)(53,150)(54,149)(55,148)(56,147)(57,146)(58,153)(59,171)(60,170)(61,169)(62,168)(63,167)(64,166)(65,165)(66,164)(67,163)(68,162)(69,161)(70,160)(71,159)(72,158)(73,157)(74,156)(75,155)(76,154)(77,183)(78,182)(79,181)(80,180)(81,179)(82,178)(83,177)(84,176)(85,175)(86,174)(87,173)(88,172)(89,190)(90,189)(91,188)(92,187)(93,186)(94,185)(95,184)>;
G:=Group( (1,137,93,115,61,96,48,187,21,170)(2,138,94,116,62,97,49,188,22,171)(3,139,95,117,63,98,50,189,23,153)(4,140,77,118,64,99,51,190,24,154)(5,141,78,119,65,100,52,172,25,155)(6,142,79,120,66,101,53,173,26,156)(7,143,80,121,67,102,54,174,27,157)(8,144,81,122,68,103,55,175,28,158)(9,145,82,123,69,104,56,176,29,159)(10,146,83,124,70,105,57,177,30,160)(11,147,84,125,71,106,39,178,31,161)(12,148,85,126,72,107,40,179,32,162)(13,149,86,127,73,108,41,180,33,163)(14,150,87,128,74,109,42,181,34,164)(15,151,88,129,75,110,43,182,35,165)(16,152,89,130,76,111,44,183,36,166)(17,134,90,131,58,112,45,184,37,167)(18,135,91,132,59,113,46,185,38,168)(19,136,92,133,60,114,47,186,20,169), (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), (1,114)(2,113)(3,112)(4,111)(5,110)(6,109)(7,108)(8,107)(9,106)(10,105)(11,104)(12,103)(13,102)(14,101)(15,100)(16,99)(17,98)(18,97)(19,96)(20,115)(21,133)(22,132)(23,131)(24,130)(25,129)(26,128)(27,127)(28,126)(29,125)(30,124)(31,123)(32,122)(33,121)(34,120)(35,119)(36,118)(37,117)(38,116)(39,145)(40,144)(41,143)(42,142)(43,141)(44,140)(45,139)(46,138)(47,137)(48,136)(49,135)(50,134)(51,152)(52,151)(53,150)(54,149)(55,148)(56,147)(57,146)(58,153)(59,171)(60,170)(61,169)(62,168)(63,167)(64,166)(65,165)(66,164)(67,163)(68,162)(69,161)(70,160)(71,159)(72,158)(73,157)(74,156)(75,155)(76,154)(77,183)(78,182)(79,181)(80,180)(81,179)(82,178)(83,177)(84,176)(85,175)(86,174)(87,173)(88,172)(89,190)(90,189)(91,188)(92,187)(93,186)(94,185)(95,184) );
G=PermutationGroup([[(1,137,93,115,61,96,48,187,21,170),(2,138,94,116,62,97,49,188,22,171),(3,139,95,117,63,98,50,189,23,153),(4,140,77,118,64,99,51,190,24,154),(5,141,78,119,65,100,52,172,25,155),(6,142,79,120,66,101,53,173,26,156),(7,143,80,121,67,102,54,174,27,157),(8,144,81,122,68,103,55,175,28,158),(9,145,82,123,69,104,56,176,29,159),(10,146,83,124,70,105,57,177,30,160),(11,147,84,125,71,106,39,178,31,161),(12,148,85,126,72,107,40,179,32,162),(13,149,86,127,73,108,41,180,33,163),(14,150,87,128,74,109,42,181,34,164),(15,151,88,129,75,110,43,182,35,165),(16,152,89,130,76,111,44,183,36,166),(17,134,90,131,58,112,45,184,37,167),(18,135,91,132,59,113,46,185,38,168),(19,136,92,133,60,114,47,186,20,169)], [(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)], [(1,114),(2,113),(3,112),(4,111),(5,110),(6,109),(7,108),(8,107),(9,106),(10,105),(11,104),(12,103),(13,102),(14,101),(15,100),(16,99),(17,98),(18,97),(19,96),(20,115),(21,133),(22,132),(23,131),(24,130),(25,129),(26,128),(27,127),(28,126),(29,125),(30,124),(31,123),(32,122),(33,121),(34,120),(35,119),(36,118),(37,117),(38,116),(39,145),(40,144),(41,143),(42,142),(43,141),(44,140),(45,139),(46,138),(47,137),(48,136),(49,135),(50,134),(51,152),(52,151),(53,150),(54,149),(55,148),(56,147),(57,146),(58,153),(59,171),(60,170),(61,169),(62,168),(63,167),(64,166),(65,165),(66,164),(67,163),(68,162),(69,161),(70,160),(71,159),(72,158),(73,157),(74,156),(75,155),(76,154),(77,183),(78,182),(79,181),(80,180),(81,179),(82,178),(83,177),(84,176),(85,175),(86,174),(87,173),(88,172),(89,190),(90,189),(91,188),(92,187),(93,186),(94,185),(95,184)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 19A | ··· | 19I | 38A | ··· | 38I | 95A | ··· | 95AJ | 190A | ··· | 190AJ |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 19 | ··· | 19 | 38 | ··· | 38 | 95 | ··· | 95 | 190 | ··· | 190 |
| size | 1 | 1 | 19 | 19 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 19 | ··· | 19 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | D19 | D38 | C5×D19 | C10×D19 |
| kernel | C10×D19 | C5×D19 | C190 | D38 | D19 | C38 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 4 | 8 | 4 | 9 | 9 | 36 | 36 |
Matrix representation of C10×D19 ►in GL2(𝔽191) generated by
| 142 | 0 |
| 0 | 142 |
| 77 | 159 |
| 190 | 55 |
| 114 | 184 |
| 1 | 77 |
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