direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C19×D13, C13⋊C38, C247⋊3C2, SmallGroup(494,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C19×D13 |
Generators and relations for C19×D13
G = < a,b,c | a19=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C19×D13
►On 247 points
Generators in S247
(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)(191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209)(210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228)(229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247)
(1 103 155 212 89 121 38 230 135 185 64 45 209)(2 104 156 213 90 122 20 231 136 186 65 46 191)(3 105 157 214 91 123 21 232 137 187 66 47 192)(4 106 158 215 92 124 22 233 138 188 67 48 193)(5 107 159 216 93 125 23 234 139 189 68 49 194)(6 108 160 217 94 126 24 235 140 190 69 50 195)(7 109 161 218 95 127 25 236 141 172 70 51 196)(8 110 162 219 77 128 26 237 142 173 71 52 197)(9 111 163 220 78 129 27 238 143 174 72 53 198)(10 112 164 221 79 130 28 239 144 175 73 54 199)(11 113 165 222 80 131 29 240 145 176 74 55 200)(12 114 166 223 81 132 30 241 146 177 75 56 201)(13 96 167 224 82 133 31 242 147 178 76 57 202)(14 97 168 225 83 115 32 243 148 179 58 39 203)(15 98 169 226 84 116 33 244 149 180 59 40 204)(16 99 170 227 85 117 34 245 150 181 60 41 205)(17 100 171 228 86 118 35 246 151 182 61 42 206)(18 101 153 210 87 119 36 247 152 183 62 43 207)(19 102 154 211 88 120 37 229 134 184 63 44 208)
(1 209)(2 191)(3 192)(4 193)(5 194)(6 195)(7 196)(8 197)(9 198)(10 199)(11 200)(12 201)(13 202)(14 203)(15 204)(16 205)(17 206)(18 207)(19 208)(39 97)(40 98)(41 99)(42 100)(43 101)(44 102)(45 103)(46 104)(47 105)(48 106)(49 107)(50 108)(51 109)(52 110)(53 111)(54 112)(55 113)(56 114)(57 96)(58 168)(59 169)(60 170)(61 171)(62 153)(63 154)(64 155)(65 156)(66 157)(67 158)(68 159)(69 160)(70 161)(71 162)(72 163)(73 164)(74 165)(75 166)(76 167)(77 142)(78 143)(79 144)(80 145)(81 146)(82 147)(83 148)(84 149)(85 150)(86 151)(87 152)(88 134)(89 135)(90 136)(91 137)(92 138)(93 139)(94 140)(95 141)(115 243)(116 244)(117 245)(118 246)(119 247)(120 229)(121 230)(122 231)(123 232)(124 233)(125 234)(126 235)(127 236)(128 237)(129 238)(130 239)(131 240)(132 241)(133 242)(172 218)(173 219)(174 220)(175 221)(176 222)(177 223)(178 224)(179 225)(180 226)(181 227)(182 228)(183 210)(184 211)(185 212)(186 213)(187 214)(188 215)(189 216)(190 217)
G:=sub<Sym(247)| (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)(191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209)(210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228)(229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247), (1,103,155,212,89,121,38,230,135,185,64,45,209)(2,104,156,213,90,122,20,231,136,186,65,46,191)(3,105,157,214,91,123,21,232,137,187,66,47,192)(4,106,158,215,92,124,22,233,138,188,67,48,193)(5,107,159,216,93,125,23,234,139,189,68,49,194)(6,108,160,217,94,126,24,235,140,190,69,50,195)(7,109,161,218,95,127,25,236,141,172,70,51,196)(8,110,162,219,77,128,26,237,142,173,71,52,197)(9,111,163,220,78,129,27,238,143,174,72,53,198)(10,112,164,221,79,130,28,239,144,175,73,54,199)(11,113,165,222,80,131,29,240,145,176,74,55,200)(12,114,166,223,81,132,30,241,146,177,75,56,201)(13,96,167,224,82,133,31,242,147,178,76,57,202)(14,97,168,225,83,115,32,243,148,179,58,39,203)(15,98,169,226,84,116,33,244,149,180,59,40,204)(16,99,170,227,85,117,34,245,150,181,60,41,205)(17,100,171,228,86,118,35,246,151,182,61,42,206)(18,101,153,210,87,119,36,247,152,183,62,43,207)(19,102,154,211,88,120,37,229,134,184,63,44,208), (1,209)(2,191)(3,192)(4,193)(5,194)(6,195)(7,196)(8,197)(9,198)(10,199)(11,200)(12,201)(13,202)(14,203)(15,204)(16,205)(17,206)(18,207)(19,208)(39,97)(40,98)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104)(47,105)(48,106)(49,107)(50,108)(51,109)(52,110)(53,111)(54,112)(55,113)(56,114)(57,96)(58,168)(59,169)(60,170)(61,171)(62,153)(63,154)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,161)(71,162)(72,163)(73,164)(74,165)(75,166)(76,167)(77,142)(78,143)(79,144)(80,145)(81,146)(82,147)(83,148)(84,149)(85,150)(86,151)(87,152)(88,134)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,141)(115,243)(116,244)(117,245)(118,246)(119,247)(120,229)(121,230)(122,231)(123,232)(124,233)(125,234)(126,235)(127,236)(128,237)(129,238)(130,239)(131,240)(132,241)(133,242)(172,218)(173,219)(174,220)(175,221)(176,222)(177,223)(178,224)(179,225)(180,226)(181,227)(182,228)(183,210)(184,211)(185,212)(186,213)(187,214)(188,215)(189,216)(190,217)>;
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)(191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209)(210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228)(229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247), (1,103,155,212,89,121,38,230,135,185,64,45,209)(2,104,156,213,90,122,20,231,136,186,65,46,191)(3,105,157,214,91,123,21,232,137,187,66,47,192)(4,106,158,215,92,124,22,233,138,188,67,48,193)(5,107,159,216,93,125,23,234,139,189,68,49,194)(6,108,160,217,94,126,24,235,140,190,69,50,195)(7,109,161,218,95,127,25,236,141,172,70,51,196)(8,110,162,219,77,128,26,237,142,173,71,52,197)(9,111,163,220,78,129,27,238,143,174,72,53,198)(10,112,164,221,79,130,28,239,144,175,73,54,199)(11,113,165,222,80,131,29,240,145,176,74,55,200)(12,114,166,223,81,132,30,241,146,177,75,56,201)(13,96,167,224,82,133,31,242,147,178,76,57,202)(14,97,168,225,83,115,32,243,148,179,58,39,203)(15,98,169,226,84,116,33,244,149,180,59,40,204)(16,99,170,227,85,117,34,245,150,181,60,41,205)(17,100,171,228,86,118,35,246,151,182,61,42,206)(18,101,153,210,87,119,36,247,152,183,62,43,207)(19,102,154,211,88,120,37,229,134,184,63,44,208), (1,209)(2,191)(3,192)(4,193)(5,194)(6,195)(7,196)(8,197)(9,198)(10,199)(11,200)(12,201)(13,202)(14,203)(15,204)(16,205)(17,206)(18,207)(19,208)(39,97)(40,98)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104)(47,105)(48,106)(49,107)(50,108)(51,109)(52,110)(53,111)(54,112)(55,113)(56,114)(57,96)(58,168)(59,169)(60,170)(61,171)(62,153)(63,154)(64,155)(65,156)(66,157)(67,158)(68,159)(69,160)(70,161)(71,162)(72,163)(73,164)(74,165)(75,166)(76,167)(77,142)(78,143)(79,144)(80,145)(81,146)(82,147)(83,148)(84,149)(85,150)(86,151)(87,152)(88,134)(89,135)(90,136)(91,137)(92,138)(93,139)(94,140)(95,141)(115,243)(116,244)(117,245)(118,246)(119,247)(120,229)(121,230)(122,231)(123,232)(124,233)(125,234)(126,235)(127,236)(128,237)(129,238)(130,239)(131,240)(132,241)(133,242)(172,218)(173,219)(174,220)(175,221)(176,222)(177,223)(178,224)(179,225)(180,226)(181,227)(182,228)(183,210)(184,211)(185,212)(186,213)(187,214)(188,215)(189,216)(190,217) );
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),(191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209),(210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228),(229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247)], [(1,103,155,212,89,121,38,230,135,185,64,45,209),(2,104,156,213,90,122,20,231,136,186,65,46,191),(3,105,157,214,91,123,21,232,137,187,66,47,192),(4,106,158,215,92,124,22,233,138,188,67,48,193),(5,107,159,216,93,125,23,234,139,189,68,49,194),(6,108,160,217,94,126,24,235,140,190,69,50,195),(7,109,161,218,95,127,25,236,141,172,70,51,196),(8,110,162,219,77,128,26,237,142,173,71,52,197),(9,111,163,220,78,129,27,238,143,174,72,53,198),(10,112,164,221,79,130,28,239,144,175,73,54,199),(11,113,165,222,80,131,29,240,145,176,74,55,200),(12,114,166,223,81,132,30,241,146,177,75,56,201),(13,96,167,224,82,133,31,242,147,178,76,57,202),(14,97,168,225,83,115,32,243,148,179,58,39,203),(15,98,169,226,84,116,33,244,149,180,59,40,204),(16,99,170,227,85,117,34,245,150,181,60,41,205),(17,100,171,228,86,118,35,246,151,182,61,42,206),(18,101,153,210,87,119,36,247,152,183,62,43,207),(19,102,154,211,88,120,37,229,134,184,63,44,208)], [(1,209),(2,191),(3,192),(4,193),(5,194),(6,195),(7,196),(8,197),(9,198),(10,199),(11,200),(12,201),(13,202),(14,203),(15,204),(16,205),(17,206),(18,207),(19,208),(39,97),(40,98),(41,99),(42,100),(43,101),(44,102),(45,103),(46,104),(47,105),(48,106),(49,107),(50,108),(51,109),(52,110),(53,111),(54,112),(55,113),(56,114),(57,96),(58,168),(59,169),(60,170),(61,171),(62,153),(63,154),(64,155),(65,156),(66,157),(67,158),(68,159),(69,160),(70,161),(71,162),(72,163),(73,164),(74,165),(75,166),(76,167),(77,142),(78,143),(79,144),(80,145),(81,146),(82,147),(83,148),(84,149),(85,150),(86,151),(87,152),(88,134),(89,135),(90,136),(91,137),(92,138),(93,139),(94,140),(95,141),(115,243),(116,244),(117,245),(118,246),(119,247),(120,229),(121,230),(122,231),(123,232),(124,233),(125,234),(126,235),(127,236),(128,237),(129,238),(130,239),(131,240),(132,241),(133,242),(172,218),(173,219),(174,220),(175,221),(176,222),(177,223),(178,224),(179,225),(180,226),(181,227),(182,228),(183,210),(184,211),(185,212),(186,213),(187,214),(188,215),(189,216),(190,217)]])
152 conjugacy classes
| class | 1 | 2 | 13A | ··· | 13F | 19A | ··· | 19R | 38A | ··· | 38R | 247A | ··· | 247DD |
| order | 1 | 2 | 13 | ··· | 13 | 19 | ··· | 19 | 38 | ··· | 38 | 247 | ··· | 247 |
| size | 1 | 13 | 2 | ··· | 2 | 1 | ··· | 1 | 13 | ··· | 13 | 2 | ··· | 2 |
152 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C19 | C38 | D13 | C19×D13 |
| kernel | C19×D13 | C247 | D13 | C13 | C19 | C1 |
| # reps | 1 | 1 | 18 | 18 | 6 | 108 |
Matrix representation of C19×D13 ►in GL2(𝔽1483) generated by
| 792 | 0 |
| 0 | 792 |
| 11 | 1 |
| 238 | 561 |
| 560 | 237 |
| 967 | 923 |
G:=sub<GL(2,GF(1483))| [792,0,0,792],[11,238,1,561],[560,967,237,923] >;
C19×D13 in GAP, Magma, Sage, TeX
C_{19}\times D_{13} % in TeX
G:=Group("C19xD13"); // GroupNames label
G:=SmallGroup(494,1);
// by ID
G=gap.SmallGroup(494,1);
# by ID
G:=PCGroup([3,-2,-19,-13,4106]);
// Polycyclic
G:=Group<a,b,c|a^19=b^13=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export