direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C2×C19⋊2C9, C38⋊2C9, C19⋊4C18, C114.C3, C57.2C6, C6.(C19⋊C3), C3.(C2×C19⋊C3), SmallGroup(342,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C2×C19⋊2C9 |
Generators and relations for C2×C19⋊2C9
G = < a,b,c | a2=b19=c9=1, ab=ba, ac=ca, cbc-1=b11 >
Smallest permutation representation of C2×C19⋊2C9
►Regular action on 342 points
Generators in S342
(1 172)(2 173)(3 174)(4 175)(5 176)(6 177)(7 178)(8 179)(9 180)(10 181)(11 182)(12 183)(13 184)(14 185)(15 186)(16 187)(17 188)(18 189)(19 190)(20 191)(21 192)(22 193)(23 194)(24 195)(25 196)(26 197)(27 198)(28 199)(29 200)(30 201)(31 202)(32 203)(33 204)(34 205)(35 206)(36 207)(37 208)(38 209)(39 210)(40 211)(41 212)(42 213)(43 214)(44 215)(45 216)(46 217)(47 218)(48 219)(49 220)(50 221)(51 222)(52 223)(53 224)(54 225)(55 226)(56 227)(57 228)(58 245)(59 246)(60 247)(61 229)(62 230)(63 231)(64 232)(65 233)(66 234)(67 235)(68 236)(69 237)(70 238)(71 239)(72 240)(73 241)(74 242)(75 243)(76 244)(77 255)(78 256)(79 257)(80 258)(81 259)(82 260)(83 261)(84 262)(85 263)(86 264)(87 265)(88 266)(89 248)(90 249)(91 250)(92 251)(93 252)(94 253)(95 254)(96 272)(97 273)(98 274)(99 275)(100 276)(101 277)(102 278)(103 279)(104 280)(105 281)(106 282)(107 283)(108 284)(109 285)(110 267)(111 268)(112 269)(113 270)(114 271)(115 289)(116 290)(117 291)(118 292)(119 293)(120 294)(121 295)(122 296)(123 297)(124 298)(125 299)(126 300)(127 301)(128 302)(129 303)(130 304)(131 286)(132 287)(133 288)(134 322)(135 323)(136 305)(137 306)(138 307)(139 308)(140 309)(141 310)(142 311)(143 312)(144 313)(145 314)(146 315)(147 316)(148 317)(149 318)(150 319)(151 320)(152 321)(153 341)(154 342)(155 324)(156 325)(157 326)(158 327)(159 328)(160 329)(161 330)(162 331)(163 332)(164 333)(165 334)(166 335)(167 336)(168 337)(169 338)(170 339)(171 340)
(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)(248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266)(267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285)(286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304)(305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323)(324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342)
(1 155 99 39 152 82 20 124 67)(2 162 110 40 140 93 21 131 59)(3 169 102 41 147 85 22 119 70)(4 157 113 42 135 77 23 126 62)(5 164 105 43 142 88 24 133 73)(6 171 97 44 149 80 25 121 65)(7 159 108 45 137 91 26 128 76)(8 166 100 46 144 83 27 116 68)(9 154 111 47 151 94 28 123 60)(10 161 103 48 139 86 29 130 71)(11 168 114 49 146 78 30 118 63)(12 156 106 50 134 89 31 125 74)(13 163 98 51 141 81 32 132 66)(14 170 109 52 148 92 33 120 58)(15 158 101 53 136 84 34 127 69)(16 165 112 54 143 95 35 115 61)(17 153 104 55 150 87 36 122 72)(18 160 96 56 138 79 37 129 64)(19 167 107 57 145 90 38 117 75)(172 324 275 210 321 260 191 298 235)(173 331 267 211 309 252 192 286 246)(174 338 278 212 316 263 193 293 238)(175 326 270 213 323 255 194 300 230)(176 333 281 214 311 266 195 288 241)(177 340 273 215 318 258 196 295 233)(178 328 284 216 306 250 197 302 244)(179 335 276 217 313 261 198 290 236)(180 342 268 218 320 253 199 297 247)(181 330 279 219 308 264 200 304 239)(182 337 271 220 315 256 201 292 231)(183 325 282 221 322 248 202 299 242)(184 332 274 222 310 259 203 287 234)(185 339 285 223 317 251 204 294 245)(186 327 277 224 305 262 205 301 237)(187 334 269 225 312 254 206 289 229)(188 341 280 226 319 265 207 296 240)(189 329 272 227 307 257 208 303 232)(190 336 283 228 314 249 209 291 243)
G:=sub<Sym(342)| (1,172)(2,173)(3,174)(4,175)(5,176)(6,177)(7,178)(8,179)(9,180)(10,181)(11,182)(12,183)(13,184)(14,185)(15,186)(16,187)(17,188)(18,189)(19,190)(20,191)(21,192)(22,193)(23,194)(24,195)(25,196)(26,197)(27,198)(28,199)(29,200)(30,201)(31,202)(32,203)(33,204)(34,205)(35,206)(36,207)(37,208)(38,209)(39,210)(40,211)(41,212)(42,213)(43,214)(44,215)(45,216)(46,217)(47,218)(48,219)(49,220)(50,221)(51,222)(52,223)(53,224)(54,225)(55,226)(56,227)(57,228)(58,245)(59,246)(60,247)(61,229)(62,230)(63,231)(64,232)(65,233)(66,234)(67,235)(68,236)(69,237)(70,238)(71,239)(72,240)(73,241)(74,242)(75,243)(76,244)(77,255)(78,256)(79,257)(80,258)(81,259)(82,260)(83,261)(84,262)(85,263)(86,264)(87,265)(88,266)(89,248)(90,249)(91,250)(92,251)(93,252)(94,253)(95,254)(96,272)(97,273)(98,274)(99,275)(100,276)(101,277)(102,278)(103,279)(104,280)(105,281)(106,282)(107,283)(108,284)(109,285)(110,267)(111,268)(112,269)(113,270)(114,271)(115,289)(116,290)(117,291)(118,292)(119,293)(120,294)(121,295)(122,296)(123,297)(124,298)(125,299)(126,300)(127,301)(128,302)(129,303)(130,304)(131,286)(132,287)(133,288)(134,322)(135,323)(136,305)(137,306)(138,307)(139,308)(140,309)(141,310)(142,311)(143,312)(144,313)(145,314)(146,315)(147,316)(148,317)(149,318)(150,319)(151,320)(152,321)(153,341)(154,342)(155,324)(156,325)(157,326)(158,327)(159,328)(160,329)(161,330)(162,331)(163,332)(164,333)(165,334)(166,335)(167,336)(168,337)(169,338)(170,339)(171,340), (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)(248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266)(267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285)(286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304)(305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323)(324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342), (1,155,99,39,152,82,20,124,67)(2,162,110,40,140,93,21,131,59)(3,169,102,41,147,85,22,119,70)(4,157,113,42,135,77,23,126,62)(5,164,105,43,142,88,24,133,73)(6,171,97,44,149,80,25,121,65)(7,159,108,45,137,91,26,128,76)(8,166,100,46,144,83,27,116,68)(9,154,111,47,151,94,28,123,60)(10,161,103,48,139,86,29,130,71)(11,168,114,49,146,78,30,118,63)(12,156,106,50,134,89,31,125,74)(13,163,98,51,141,81,32,132,66)(14,170,109,52,148,92,33,120,58)(15,158,101,53,136,84,34,127,69)(16,165,112,54,143,95,35,115,61)(17,153,104,55,150,87,36,122,72)(18,160,96,56,138,79,37,129,64)(19,167,107,57,145,90,38,117,75)(172,324,275,210,321,260,191,298,235)(173,331,267,211,309,252,192,286,246)(174,338,278,212,316,263,193,293,238)(175,326,270,213,323,255,194,300,230)(176,333,281,214,311,266,195,288,241)(177,340,273,215,318,258,196,295,233)(178,328,284,216,306,250,197,302,244)(179,335,276,217,313,261,198,290,236)(180,342,268,218,320,253,199,297,247)(181,330,279,219,308,264,200,304,239)(182,337,271,220,315,256,201,292,231)(183,325,282,221,322,248,202,299,242)(184,332,274,222,310,259,203,287,234)(185,339,285,223,317,251,204,294,245)(186,327,277,224,305,262,205,301,237)(187,334,269,225,312,254,206,289,229)(188,341,280,226,319,265,207,296,240)(189,329,272,227,307,257,208,303,232)(190,336,283,228,314,249,209,291,243)>;
G:=Group( (1,172)(2,173)(3,174)(4,175)(5,176)(6,177)(7,178)(8,179)(9,180)(10,181)(11,182)(12,183)(13,184)(14,185)(15,186)(16,187)(17,188)(18,189)(19,190)(20,191)(21,192)(22,193)(23,194)(24,195)(25,196)(26,197)(27,198)(28,199)(29,200)(30,201)(31,202)(32,203)(33,204)(34,205)(35,206)(36,207)(37,208)(38,209)(39,210)(40,211)(41,212)(42,213)(43,214)(44,215)(45,216)(46,217)(47,218)(48,219)(49,220)(50,221)(51,222)(52,223)(53,224)(54,225)(55,226)(56,227)(57,228)(58,245)(59,246)(60,247)(61,229)(62,230)(63,231)(64,232)(65,233)(66,234)(67,235)(68,236)(69,237)(70,238)(71,239)(72,240)(73,241)(74,242)(75,243)(76,244)(77,255)(78,256)(79,257)(80,258)(81,259)(82,260)(83,261)(84,262)(85,263)(86,264)(87,265)(88,266)(89,248)(90,249)(91,250)(92,251)(93,252)(94,253)(95,254)(96,272)(97,273)(98,274)(99,275)(100,276)(101,277)(102,278)(103,279)(104,280)(105,281)(106,282)(107,283)(108,284)(109,285)(110,267)(111,268)(112,269)(113,270)(114,271)(115,289)(116,290)(117,291)(118,292)(119,293)(120,294)(121,295)(122,296)(123,297)(124,298)(125,299)(126,300)(127,301)(128,302)(129,303)(130,304)(131,286)(132,287)(133,288)(134,322)(135,323)(136,305)(137,306)(138,307)(139,308)(140,309)(141,310)(142,311)(143,312)(144,313)(145,314)(146,315)(147,316)(148,317)(149,318)(150,319)(151,320)(152,321)(153,341)(154,342)(155,324)(156,325)(157,326)(158,327)(159,328)(160,329)(161,330)(162,331)(163,332)(164,333)(165,334)(166,335)(167,336)(168,337)(169,338)(170,339)(171,340), (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)(248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266)(267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285)(286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304)(305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323)(324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342), (1,155,99,39,152,82,20,124,67)(2,162,110,40,140,93,21,131,59)(3,169,102,41,147,85,22,119,70)(4,157,113,42,135,77,23,126,62)(5,164,105,43,142,88,24,133,73)(6,171,97,44,149,80,25,121,65)(7,159,108,45,137,91,26,128,76)(8,166,100,46,144,83,27,116,68)(9,154,111,47,151,94,28,123,60)(10,161,103,48,139,86,29,130,71)(11,168,114,49,146,78,30,118,63)(12,156,106,50,134,89,31,125,74)(13,163,98,51,141,81,32,132,66)(14,170,109,52,148,92,33,120,58)(15,158,101,53,136,84,34,127,69)(16,165,112,54,143,95,35,115,61)(17,153,104,55,150,87,36,122,72)(18,160,96,56,138,79,37,129,64)(19,167,107,57,145,90,38,117,75)(172,324,275,210,321,260,191,298,235)(173,331,267,211,309,252,192,286,246)(174,338,278,212,316,263,193,293,238)(175,326,270,213,323,255,194,300,230)(176,333,281,214,311,266,195,288,241)(177,340,273,215,318,258,196,295,233)(178,328,284,216,306,250,197,302,244)(179,335,276,217,313,261,198,290,236)(180,342,268,218,320,253,199,297,247)(181,330,279,219,308,264,200,304,239)(182,337,271,220,315,256,201,292,231)(183,325,282,221,322,248,202,299,242)(184,332,274,222,310,259,203,287,234)(185,339,285,223,317,251,204,294,245)(186,327,277,224,305,262,205,301,237)(187,334,269,225,312,254,206,289,229)(188,341,280,226,319,265,207,296,240)(189,329,272,227,307,257,208,303,232)(190,336,283,228,314,249,209,291,243) );
G=PermutationGroup([[(1,172),(2,173),(3,174),(4,175),(5,176),(6,177),(7,178),(8,179),(9,180),(10,181),(11,182),(12,183),(13,184),(14,185),(15,186),(16,187),(17,188),(18,189),(19,190),(20,191),(21,192),(22,193),(23,194),(24,195),(25,196),(26,197),(27,198),(28,199),(29,200),(30,201),(31,202),(32,203),(33,204),(34,205),(35,206),(36,207),(37,208),(38,209),(39,210),(40,211),(41,212),(42,213),(43,214),(44,215),(45,216),(46,217),(47,218),(48,219),(49,220),(50,221),(51,222),(52,223),(53,224),(54,225),(55,226),(56,227),(57,228),(58,245),(59,246),(60,247),(61,229),(62,230),(63,231),(64,232),(65,233),(66,234),(67,235),(68,236),(69,237),(70,238),(71,239),(72,240),(73,241),(74,242),(75,243),(76,244),(77,255),(78,256),(79,257),(80,258),(81,259),(82,260),(83,261),(84,262),(85,263),(86,264),(87,265),(88,266),(89,248),(90,249),(91,250),(92,251),(93,252),(94,253),(95,254),(96,272),(97,273),(98,274),(99,275),(100,276),(101,277),(102,278),(103,279),(104,280),(105,281),(106,282),(107,283),(108,284),(109,285),(110,267),(111,268),(112,269),(113,270),(114,271),(115,289),(116,290),(117,291),(118,292),(119,293),(120,294),(121,295),(122,296),(123,297),(124,298),(125,299),(126,300),(127,301),(128,302),(129,303),(130,304),(131,286),(132,287),(133,288),(134,322),(135,323),(136,305),(137,306),(138,307),(139,308),(140,309),(141,310),(142,311),(143,312),(144,313),(145,314),(146,315),(147,316),(148,317),(149,318),(150,319),(151,320),(152,321),(153,341),(154,342),(155,324),(156,325),(157,326),(158,327),(159,328),(160,329),(161,330),(162,331),(163,332),(164,333),(165,334),(166,335),(167,336),(168,337),(169,338),(170,339),(171,340)], [(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),(248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266),(267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285),(286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304),(305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323),(324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342)], [(1,155,99,39,152,82,20,124,67),(2,162,110,40,140,93,21,131,59),(3,169,102,41,147,85,22,119,70),(4,157,113,42,135,77,23,126,62),(5,164,105,43,142,88,24,133,73),(6,171,97,44,149,80,25,121,65),(7,159,108,45,137,91,26,128,76),(8,166,100,46,144,83,27,116,68),(9,154,111,47,151,94,28,123,60),(10,161,103,48,139,86,29,130,71),(11,168,114,49,146,78,30,118,63),(12,156,106,50,134,89,31,125,74),(13,163,98,51,141,81,32,132,66),(14,170,109,52,148,92,33,120,58),(15,158,101,53,136,84,34,127,69),(16,165,112,54,143,95,35,115,61),(17,153,104,55,150,87,36,122,72),(18,160,96,56,138,79,37,129,64),(19,167,107,57,145,90,38,117,75),(172,324,275,210,321,260,191,298,235),(173,331,267,211,309,252,192,286,246),(174,338,278,212,316,263,193,293,238),(175,326,270,213,323,255,194,300,230),(176,333,281,214,311,266,195,288,241),(177,340,273,215,318,258,196,295,233),(178,328,284,216,306,250,197,302,244),(179,335,276,217,313,261,198,290,236),(180,342,268,218,320,253,199,297,247),(181,330,279,219,308,264,200,304,239),(182,337,271,220,315,256,201,292,231),(183,325,282,221,322,248,202,299,242),(184,332,274,222,310,259,203,287,234),(185,339,285,223,317,251,204,294,245),(186,327,277,224,305,262,205,301,237),(187,334,269,225,312,254,206,289,229),(188,341,280,226,319,265,207,296,240),(189,329,272,227,307,257,208,303,232),(190,336,283,228,314,249,209,291,243)]])
54 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 9A | ··· | 9F | 18A | ··· | 18F | 19A | ··· | 19F | 38A | ··· | 38F | 57A | ··· | 57L | 114A | ··· | 114L |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 9 | ··· | 9 | 18 | ··· | 18 | 19 | ··· | 19 | 38 | ··· | 38 | 57 | ··· | 57 | 114 | ··· | 114 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 19 | ··· | 19 | 19 | ··· | 19 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 | 3 | ··· | 3 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | C19⋊C3 | C2×C19⋊C3 | C19⋊2C9 | C2×C19⋊2C9 |
| kernel | C2×C19⋊2C9 | C19⋊2C9 | C114 | C57 | C38 | C19 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 |
Matrix representation of C2×C19⋊2C9 ►in GL3(𝔽7) generated by
| 6 | 0 | 0 |
| 0 | 6 | 0 |
| 0 | 0 | 6 |
| 4 | 2 | 2 |
| 6 | 6 | 3 |
| 2 | 6 | 4 |
| 0 | 4 | 3 |
| 6 | 3 | 4 |
| 0 | 4 | 4 |
G:=sub<GL(3,GF(7))| [6,0,0,0,6,0,0,0,6],[4,6,2,2,6,6,2,3,4],[0,6,0,4,3,4,3,4,4] >;
C2×C19⋊2C9 in GAP, Magma, Sage, TeX
C_2\times C_{19}\rtimes_2C_9 % in TeX
G:=Group("C2xC19:2C9"); // GroupNames label
G:=SmallGroup(342,2);
// by ID
G=gap.SmallGroup(342,2);
# by ID
G:=PCGroup([4,-2,-3,-3,-19,29,1015]);
// Polycyclic
G:=Group<a,b,c|a^2=b^19=c^9=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^11>;
// generators/relations
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