direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C13×D19, C19⋊C26, C247⋊2C2, SmallGroup(494,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C13×D19 |
Generators and relations for C13×D19
G = < a,b,c | a13=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C13×D19
►On 247 points
Generators in S247
(1 246 226 192 189 160 149 129 97 83 58 51 20)(2 247 227 193 190 161 150 130 98 84 59 52 21)(3 229 228 194 172 162 151 131 99 85 60 53 22)(4 230 210 195 173 163 152 132 100 86 61 54 23)(5 231 211 196 174 164 134 133 101 87 62 55 24)(6 232 212 197 175 165 135 115 102 88 63 56 25)(7 233 213 198 176 166 136 116 103 89 64 57 26)(8 234 214 199 177 167 137 117 104 90 65 39 27)(9 235 215 200 178 168 138 118 105 91 66 40 28)(10 236 216 201 179 169 139 119 106 92 67 41 29)(11 237 217 202 180 170 140 120 107 93 68 42 30)(12 238 218 203 181 171 141 121 108 94 69 43 31)(13 239 219 204 182 153 142 122 109 95 70 44 32)(14 240 220 205 183 154 143 123 110 77 71 45 33)(15 241 221 206 184 155 144 124 111 78 72 46 34)(16 242 222 207 185 156 145 125 112 79 73 47 35)(17 243 223 208 186 157 146 126 113 80 74 48 36)(18 244 224 209 187 158 147 127 114 81 75 49 37)(19 245 225 191 188 159 148 128 96 82 76 50 38)
(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 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 38)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)(39 43)(40 42)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(58 76)(59 75)(60 74)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(77 88)(78 87)(79 86)(80 85)(81 84)(82 83)(89 95)(90 94)(91 93)(96 97)(98 114)(99 113)(100 112)(101 111)(102 110)(103 109)(104 108)(105 107)(115 123)(116 122)(117 121)(118 120)(124 133)(125 132)(126 131)(127 130)(128 129)(134 144)(135 143)(136 142)(137 141)(138 140)(145 152)(146 151)(147 150)(148 149)(153 166)(154 165)(155 164)(156 163)(157 162)(158 161)(159 160)(167 171)(168 170)(172 186)(173 185)(174 184)(175 183)(176 182)(177 181)(178 180)(187 190)(188 189)(191 192)(193 209)(194 208)(195 207)(196 206)(197 205)(198 204)(199 203)(200 202)(210 222)(211 221)(212 220)(213 219)(214 218)(215 217)(223 228)(224 227)(225 226)(229 243)(230 242)(231 241)(232 240)(233 239)(234 238)(235 237)(244 247)(245 246)
G:=sub<Sym(247)| (1,246,226,192,189,160,149,129,97,83,58,51,20)(2,247,227,193,190,161,150,130,98,84,59,52,21)(3,229,228,194,172,162,151,131,99,85,60,53,22)(4,230,210,195,173,163,152,132,100,86,61,54,23)(5,231,211,196,174,164,134,133,101,87,62,55,24)(6,232,212,197,175,165,135,115,102,88,63,56,25)(7,233,213,198,176,166,136,116,103,89,64,57,26)(8,234,214,199,177,167,137,117,104,90,65,39,27)(9,235,215,200,178,168,138,118,105,91,66,40,28)(10,236,216,201,179,169,139,119,106,92,67,41,29)(11,237,217,202,180,170,140,120,107,93,68,42,30)(12,238,218,203,181,171,141,121,108,94,69,43,31)(13,239,219,204,182,153,142,122,109,95,70,44,32)(14,240,220,205,183,154,143,123,110,77,71,45,33)(15,241,221,206,184,155,144,124,111,78,72,46,34)(16,242,222,207,185,156,145,125,112,79,73,47,35)(17,243,223,208,186,157,146,126,113,80,74,48,36)(18,244,224,209,187,158,147,127,114,81,75,49,37)(19,245,225,191,188,159,148,128,96,82,76,50,38), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(39,43)(40,42)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(58,76)(59,75)(60,74)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)(89,95)(90,94)(91,93)(96,97)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(115,123)(116,122)(117,121)(118,120)(124,133)(125,132)(126,131)(127,130)(128,129)(134,144)(135,143)(136,142)(137,141)(138,140)(145,152)(146,151)(147,150)(148,149)(153,166)(154,165)(155,164)(156,163)(157,162)(158,161)(159,160)(167,171)(168,170)(172,186)(173,185)(174,184)(175,183)(176,182)(177,181)(178,180)(187,190)(188,189)(191,192)(193,209)(194,208)(195,207)(196,206)(197,205)(198,204)(199,203)(200,202)(210,222)(211,221)(212,220)(213,219)(214,218)(215,217)(223,228)(224,227)(225,226)(229,243)(230,242)(231,241)(232,240)(233,239)(234,238)(235,237)(244,247)(245,246)>;
G:=Group( (1,246,226,192,189,160,149,129,97,83,58,51,20)(2,247,227,193,190,161,150,130,98,84,59,52,21)(3,229,228,194,172,162,151,131,99,85,60,53,22)(4,230,210,195,173,163,152,132,100,86,61,54,23)(5,231,211,196,174,164,134,133,101,87,62,55,24)(6,232,212,197,175,165,135,115,102,88,63,56,25)(7,233,213,198,176,166,136,116,103,89,64,57,26)(8,234,214,199,177,167,137,117,104,90,65,39,27)(9,235,215,200,178,168,138,118,105,91,66,40,28)(10,236,216,201,179,169,139,119,106,92,67,41,29)(11,237,217,202,180,170,140,120,107,93,68,42,30)(12,238,218,203,181,171,141,121,108,94,69,43,31)(13,239,219,204,182,153,142,122,109,95,70,44,32)(14,240,220,205,183,154,143,123,110,77,71,45,33)(15,241,221,206,184,155,144,124,111,78,72,46,34)(16,242,222,207,185,156,145,125,112,79,73,47,35)(17,243,223,208,186,157,146,126,113,80,74,48,36)(18,244,224,209,187,158,147,127,114,81,75,49,37)(19,245,225,191,188,159,148,128,96,82,76,50,38), (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), (1,19)(2,18)(3,17)(4,16)(5,15)(6,14)(7,13)(8,12)(9,11)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(39,43)(40,42)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(58,76)(59,75)(60,74)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)(89,95)(90,94)(91,93)(96,97)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(115,123)(116,122)(117,121)(118,120)(124,133)(125,132)(126,131)(127,130)(128,129)(134,144)(135,143)(136,142)(137,141)(138,140)(145,152)(146,151)(147,150)(148,149)(153,166)(154,165)(155,164)(156,163)(157,162)(158,161)(159,160)(167,171)(168,170)(172,186)(173,185)(174,184)(175,183)(176,182)(177,181)(178,180)(187,190)(188,189)(191,192)(193,209)(194,208)(195,207)(196,206)(197,205)(198,204)(199,203)(200,202)(210,222)(211,221)(212,220)(213,219)(214,218)(215,217)(223,228)(224,227)(225,226)(229,243)(230,242)(231,241)(232,240)(233,239)(234,238)(235,237)(244,247)(245,246) );
G=PermutationGroup([[(1,246,226,192,189,160,149,129,97,83,58,51,20),(2,247,227,193,190,161,150,130,98,84,59,52,21),(3,229,228,194,172,162,151,131,99,85,60,53,22),(4,230,210,195,173,163,152,132,100,86,61,54,23),(5,231,211,196,174,164,134,133,101,87,62,55,24),(6,232,212,197,175,165,135,115,102,88,63,56,25),(7,233,213,198,176,166,136,116,103,89,64,57,26),(8,234,214,199,177,167,137,117,104,90,65,39,27),(9,235,215,200,178,168,138,118,105,91,66,40,28),(10,236,216,201,179,169,139,119,106,92,67,41,29),(11,237,217,202,180,170,140,120,107,93,68,42,30),(12,238,218,203,181,171,141,121,108,94,69,43,31),(13,239,219,204,182,153,142,122,109,95,70,44,32),(14,240,220,205,183,154,143,123,110,77,71,45,33),(15,241,221,206,184,155,144,124,111,78,72,46,34),(16,242,222,207,185,156,145,125,112,79,73,47,35),(17,243,223,208,186,157,146,126,113,80,74,48,36),(18,244,224,209,187,158,147,127,114,81,75,49,37),(19,245,225,191,188,159,148,128,96,82,76,50,38)], [(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)], [(1,19),(2,18),(3,17),(4,16),(5,15),(6,14),(7,13),(8,12),(9,11),(20,38),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,30),(39,43),(40,42),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(58,76),(59,75),(60,74),(61,73),(62,72),(63,71),(64,70),(65,69),(66,68),(77,88),(78,87),(79,86),(80,85),(81,84),(82,83),(89,95),(90,94),(91,93),(96,97),(98,114),(99,113),(100,112),(101,111),(102,110),(103,109),(104,108),(105,107),(115,123),(116,122),(117,121),(118,120),(124,133),(125,132),(126,131),(127,130),(128,129),(134,144),(135,143),(136,142),(137,141),(138,140),(145,152),(146,151),(147,150),(148,149),(153,166),(154,165),(155,164),(156,163),(157,162),(158,161),(159,160),(167,171),(168,170),(172,186),(173,185),(174,184),(175,183),(176,182),(177,181),(178,180),(187,190),(188,189),(191,192),(193,209),(194,208),(195,207),(196,206),(197,205),(198,204),(199,203),(200,202),(210,222),(211,221),(212,220),(213,219),(214,218),(215,217),(223,228),(224,227),(225,226),(229,243),(230,242),(231,241),(232,240),(233,239),(234,238),(235,237),(244,247),(245,246)]])
143 conjugacy classes
| class | 1 | 2 | 13A | ··· | 13L | 19A | ··· | 19I | 26A | ··· | 26L | 247A | ··· | 247DD |
| order | 1 | 2 | 13 | ··· | 13 | 19 | ··· | 19 | 26 | ··· | 26 | 247 | ··· | 247 |
| size | 1 | 19 | 1 | ··· | 1 | 2 | ··· | 2 | 19 | ··· | 19 | 2 | ··· | 2 |
143 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||
| image | C1 | C2 | C13 | C26 | D19 | C13×D19 |
| kernel | C13×D19 | C247 | D19 | C19 | C13 | C1 |
| # reps | 1 | 1 | 12 | 12 | 9 | 108 |
Matrix representation of C13×D19 ►in GL2(𝔽1483) generated by
| 1318 | 0 |
| 0 | 1318 |
| 624 | 82 |
| 1482 | 7 |
| 967 | 1116 |
| 677 | 516 |
G:=sub<GL(2,GF(1483))| [1318,0,0,1318],[624,1482,82,7],[967,677,1116,516] >;
C13×D19 in GAP, Magma, Sage, TeX
C_{13}\times D_{19} % in TeX
G:=Group("C13xD19"); // GroupNames label
G:=SmallGroup(494,2);
// by ID
G=gap.SmallGroup(494,2);
# by ID
G:=PCGroup([3,-2,-13,-19,4214]);
// Polycyclic
G:=Group<a,b,c|a^13=b^19=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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