direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×Dic34, C51⋊3Q8, C68.1C6, C204.3C2, C6.13D34, C12.3D17, Dic17.1C6, C102.13C22, C17⋊(C3×Q8), C4.(C3×D17), C34.1(C2×C6), C2.3(C6×D17), (C3×Dic17).2C2, SmallGroup(408,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Dic34
G = < a,b,c | a3=b68=1, c2=b34, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×Dic34
►Regular action on 408 points
Generators in S408
(1 71 366)(2 72 367)(3 73 368)(4 74 369)(5 75 370)(6 76 371)(7 77 372)(8 78 373)(9 79 374)(10 80 375)(11 81 376)(12 82 377)(13 83 378)(14 84 379)(15 85 380)(16 86 381)(17 87 382)(18 88 383)(19 89 384)(20 90 385)(21 91 386)(22 92 387)(23 93 388)(24 94 389)(25 95 390)(26 96 391)(27 97 392)(28 98 393)(29 99 394)(30 100 395)(31 101 396)(32 102 397)(33 103 398)(34 104 399)(35 105 400)(36 106 401)(37 107 402)(38 108 403)(39 109 404)(40 110 405)(41 111 406)(42 112 407)(43 113 408)(44 114 341)(45 115 342)(46 116 343)(47 117 344)(48 118 345)(49 119 346)(50 120 347)(51 121 348)(52 122 349)(53 123 350)(54 124 351)(55 125 352)(56 126 353)(57 127 354)(58 128 355)(59 129 356)(60 130 357)(61 131 358)(62 132 359)(63 133 360)(64 134 361)(65 135 362)(66 136 363)(67 69 364)(68 70 365)(137 262 293)(138 263 294)(139 264 295)(140 265 296)(141 266 297)(142 267 298)(143 268 299)(144 269 300)(145 270 301)(146 271 302)(147 272 303)(148 205 304)(149 206 305)(150 207 306)(151 208 307)(152 209 308)(153 210 309)(154 211 310)(155 212 311)(156 213 312)(157 214 313)(158 215 314)(159 216 315)(160 217 316)(161 218 317)(162 219 318)(163 220 319)(164 221 320)(165 222 321)(166 223 322)(167 224 323)(168 225 324)(169 226 325)(170 227 326)(171 228 327)(172 229 328)(173 230 329)(174 231 330)(175 232 331)(176 233 332)(177 234 333)(178 235 334)(179 236 335)(180 237 336)(181 238 337)(182 239 338)(183 240 339)(184 241 340)(185 242 273)(186 243 274)(187 244 275)(188 245 276)(189 246 277)(190 247 278)(191 248 279)(192 249 280)(193 250 281)(194 251 282)(195 252 283)(196 253 284)(197 254 285)(198 255 286)(199 256 287)(200 257 288)(201 258 289)(202 259 290)(203 260 291)(204 261 292)
(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)(341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408)
(1 245 35 211)(2 244 36 210)(3 243 37 209)(4 242 38 208)(5 241 39 207)(6 240 40 206)(7 239 41 205)(8 238 42 272)(9 237 43 271)(10 236 44 270)(11 235 45 269)(12 234 46 268)(13 233 47 267)(14 232 48 266)(15 231 49 265)(16 230 50 264)(17 229 51 263)(18 228 52 262)(19 227 53 261)(20 226 54 260)(21 225 55 259)(22 224 56 258)(23 223 57 257)(24 222 58 256)(25 221 59 255)(26 220 60 254)(27 219 61 253)(28 218 62 252)(29 217 63 251)(30 216 64 250)(31 215 65 249)(32 214 66 248)(33 213 67 247)(34 212 68 246)(69 278 103 312)(70 277 104 311)(71 276 105 310)(72 275 106 309)(73 274 107 308)(74 273 108 307)(75 340 109 306)(76 339 110 305)(77 338 111 304)(78 337 112 303)(79 336 113 302)(80 335 114 301)(81 334 115 300)(82 333 116 299)(83 332 117 298)(84 331 118 297)(85 330 119 296)(86 329 120 295)(87 328 121 294)(88 327 122 293)(89 326 123 292)(90 325 124 291)(91 324 125 290)(92 323 126 289)(93 322 127 288)(94 321 128 287)(95 320 129 286)(96 319 130 285)(97 318 131 284)(98 317 132 283)(99 316 133 282)(100 315 134 281)(101 314 135 280)(102 313 136 279)(137 383 171 349)(138 382 172 348)(139 381 173 347)(140 380 174 346)(141 379 175 345)(142 378 176 344)(143 377 177 343)(144 376 178 342)(145 375 179 341)(146 374 180 408)(147 373 181 407)(148 372 182 406)(149 371 183 405)(150 370 184 404)(151 369 185 403)(152 368 186 402)(153 367 187 401)(154 366 188 400)(155 365 189 399)(156 364 190 398)(157 363 191 397)(158 362 192 396)(159 361 193 395)(160 360 194 394)(161 359 195 393)(162 358 196 392)(163 357 197 391)(164 356 198 390)(165 355 199 389)(166 354 200 388)(167 353 201 387)(168 352 202 386)(169 351 203 385)(170 350 204 384)
G:=sub<Sym(408)| (1,71,366)(2,72,367)(3,73,368)(4,74,369)(5,75,370)(6,76,371)(7,77,372)(8,78,373)(9,79,374)(10,80,375)(11,81,376)(12,82,377)(13,83,378)(14,84,379)(15,85,380)(16,86,381)(17,87,382)(18,88,383)(19,89,384)(20,90,385)(21,91,386)(22,92,387)(23,93,388)(24,94,389)(25,95,390)(26,96,391)(27,97,392)(28,98,393)(29,99,394)(30,100,395)(31,101,396)(32,102,397)(33,103,398)(34,104,399)(35,105,400)(36,106,401)(37,107,402)(38,108,403)(39,109,404)(40,110,405)(41,111,406)(42,112,407)(43,113,408)(44,114,341)(45,115,342)(46,116,343)(47,117,344)(48,118,345)(49,119,346)(50,120,347)(51,121,348)(52,122,349)(53,123,350)(54,124,351)(55,125,352)(56,126,353)(57,127,354)(58,128,355)(59,129,356)(60,130,357)(61,131,358)(62,132,359)(63,133,360)(64,134,361)(65,135,362)(66,136,363)(67,69,364)(68,70,365)(137,262,293)(138,263,294)(139,264,295)(140,265,296)(141,266,297)(142,267,298)(143,268,299)(144,269,300)(145,270,301)(146,271,302)(147,272,303)(148,205,304)(149,206,305)(150,207,306)(151,208,307)(152,209,308)(153,210,309)(154,211,310)(155,212,311)(156,213,312)(157,214,313)(158,215,314)(159,216,315)(160,217,316)(161,218,317)(162,219,318)(163,220,319)(164,221,320)(165,222,321)(166,223,322)(167,224,323)(168,225,324)(169,226,325)(170,227,326)(171,228,327)(172,229,328)(173,230,329)(174,231,330)(175,232,331)(176,233,332)(177,234,333)(178,235,334)(179,236,335)(180,237,336)(181,238,337)(182,239,338)(183,240,339)(184,241,340)(185,242,273)(186,243,274)(187,244,275)(188,245,276)(189,246,277)(190,247,278)(191,248,279)(192,249,280)(193,250,281)(194,251,282)(195,252,283)(196,253,284)(197,254,285)(198,255,286)(199,256,287)(200,257,288)(201,258,289)(202,259,290)(203,260,291)(204,261,292), (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)(341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408), (1,245,35,211)(2,244,36,210)(3,243,37,209)(4,242,38,208)(5,241,39,207)(6,240,40,206)(7,239,41,205)(8,238,42,272)(9,237,43,271)(10,236,44,270)(11,235,45,269)(12,234,46,268)(13,233,47,267)(14,232,48,266)(15,231,49,265)(16,230,50,264)(17,229,51,263)(18,228,52,262)(19,227,53,261)(20,226,54,260)(21,225,55,259)(22,224,56,258)(23,223,57,257)(24,222,58,256)(25,221,59,255)(26,220,60,254)(27,219,61,253)(28,218,62,252)(29,217,63,251)(30,216,64,250)(31,215,65,249)(32,214,66,248)(33,213,67,247)(34,212,68,246)(69,278,103,312)(70,277,104,311)(71,276,105,310)(72,275,106,309)(73,274,107,308)(74,273,108,307)(75,340,109,306)(76,339,110,305)(77,338,111,304)(78,337,112,303)(79,336,113,302)(80,335,114,301)(81,334,115,300)(82,333,116,299)(83,332,117,298)(84,331,118,297)(85,330,119,296)(86,329,120,295)(87,328,121,294)(88,327,122,293)(89,326,123,292)(90,325,124,291)(91,324,125,290)(92,323,126,289)(93,322,127,288)(94,321,128,287)(95,320,129,286)(96,319,130,285)(97,318,131,284)(98,317,132,283)(99,316,133,282)(100,315,134,281)(101,314,135,280)(102,313,136,279)(137,383,171,349)(138,382,172,348)(139,381,173,347)(140,380,174,346)(141,379,175,345)(142,378,176,344)(143,377,177,343)(144,376,178,342)(145,375,179,341)(146,374,180,408)(147,373,181,407)(148,372,182,406)(149,371,183,405)(150,370,184,404)(151,369,185,403)(152,368,186,402)(153,367,187,401)(154,366,188,400)(155,365,189,399)(156,364,190,398)(157,363,191,397)(158,362,192,396)(159,361,193,395)(160,360,194,394)(161,359,195,393)(162,358,196,392)(163,357,197,391)(164,356,198,390)(165,355,199,389)(166,354,200,388)(167,353,201,387)(168,352,202,386)(169,351,203,385)(170,350,204,384)>;
G:=Group( (1,71,366)(2,72,367)(3,73,368)(4,74,369)(5,75,370)(6,76,371)(7,77,372)(8,78,373)(9,79,374)(10,80,375)(11,81,376)(12,82,377)(13,83,378)(14,84,379)(15,85,380)(16,86,381)(17,87,382)(18,88,383)(19,89,384)(20,90,385)(21,91,386)(22,92,387)(23,93,388)(24,94,389)(25,95,390)(26,96,391)(27,97,392)(28,98,393)(29,99,394)(30,100,395)(31,101,396)(32,102,397)(33,103,398)(34,104,399)(35,105,400)(36,106,401)(37,107,402)(38,108,403)(39,109,404)(40,110,405)(41,111,406)(42,112,407)(43,113,408)(44,114,341)(45,115,342)(46,116,343)(47,117,344)(48,118,345)(49,119,346)(50,120,347)(51,121,348)(52,122,349)(53,123,350)(54,124,351)(55,125,352)(56,126,353)(57,127,354)(58,128,355)(59,129,356)(60,130,357)(61,131,358)(62,132,359)(63,133,360)(64,134,361)(65,135,362)(66,136,363)(67,69,364)(68,70,365)(137,262,293)(138,263,294)(139,264,295)(140,265,296)(141,266,297)(142,267,298)(143,268,299)(144,269,300)(145,270,301)(146,271,302)(147,272,303)(148,205,304)(149,206,305)(150,207,306)(151,208,307)(152,209,308)(153,210,309)(154,211,310)(155,212,311)(156,213,312)(157,214,313)(158,215,314)(159,216,315)(160,217,316)(161,218,317)(162,219,318)(163,220,319)(164,221,320)(165,222,321)(166,223,322)(167,224,323)(168,225,324)(169,226,325)(170,227,326)(171,228,327)(172,229,328)(173,230,329)(174,231,330)(175,232,331)(176,233,332)(177,234,333)(178,235,334)(179,236,335)(180,237,336)(181,238,337)(182,239,338)(183,240,339)(184,241,340)(185,242,273)(186,243,274)(187,244,275)(188,245,276)(189,246,277)(190,247,278)(191,248,279)(192,249,280)(193,250,281)(194,251,282)(195,252,283)(196,253,284)(197,254,285)(198,255,286)(199,256,287)(200,257,288)(201,258,289)(202,259,290)(203,260,291)(204,261,292), (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)(341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408), (1,245,35,211)(2,244,36,210)(3,243,37,209)(4,242,38,208)(5,241,39,207)(6,240,40,206)(7,239,41,205)(8,238,42,272)(9,237,43,271)(10,236,44,270)(11,235,45,269)(12,234,46,268)(13,233,47,267)(14,232,48,266)(15,231,49,265)(16,230,50,264)(17,229,51,263)(18,228,52,262)(19,227,53,261)(20,226,54,260)(21,225,55,259)(22,224,56,258)(23,223,57,257)(24,222,58,256)(25,221,59,255)(26,220,60,254)(27,219,61,253)(28,218,62,252)(29,217,63,251)(30,216,64,250)(31,215,65,249)(32,214,66,248)(33,213,67,247)(34,212,68,246)(69,278,103,312)(70,277,104,311)(71,276,105,310)(72,275,106,309)(73,274,107,308)(74,273,108,307)(75,340,109,306)(76,339,110,305)(77,338,111,304)(78,337,112,303)(79,336,113,302)(80,335,114,301)(81,334,115,300)(82,333,116,299)(83,332,117,298)(84,331,118,297)(85,330,119,296)(86,329,120,295)(87,328,121,294)(88,327,122,293)(89,326,123,292)(90,325,124,291)(91,324,125,290)(92,323,126,289)(93,322,127,288)(94,321,128,287)(95,320,129,286)(96,319,130,285)(97,318,131,284)(98,317,132,283)(99,316,133,282)(100,315,134,281)(101,314,135,280)(102,313,136,279)(137,383,171,349)(138,382,172,348)(139,381,173,347)(140,380,174,346)(141,379,175,345)(142,378,176,344)(143,377,177,343)(144,376,178,342)(145,375,179,341)(146,374,180,408)(147,373,181,407)(148,372,182,406)(149,371,183,405)(150,370,184,404)(151,369,185,403)(152,368,186,402)(153,367,187,401)(154,366,188,400)(155,365,189,399)(156,364,190,398)(157,363,191,397)(158,362,192,396)(159,361,193,395)(160,360,194,394)(161,359,195,393)(162,358,196,392)(163,357,197,391)(164,356,198,390)(165,355,199,389)(166,354,200,388)(167,353,201,387)(168,352,202,386)(169,351,203,385)(170,350,204,384) );
G=PermutationGroup([[(1,71,366),(2,72,367),(3,73,368),(4,74,369),(5,75,370),(6,76,371),(7,77,372),(8,78,373),(9,79,374),(10,80,375),(11,81,376),(12,82,377),(13,83,378),(14,84,379),(15,85,380),(16,86,381),(17,87,382),(18,88,383),(19,89,384),(20,90,385),(21,91,386),(22,92,387),(23,93,388),(24,94,389),(25,95,390),(26,96,391),(27,97,392),(28,98,393),(29,99,394),(30,100,395),(31,101,396),(32,102,397),(33,103,398),(34,104,399),(35,105,400),(36,106,401),(37,107,402),(38,108,403),(39,109,404),(40,110,405),(41,111,406),(42,112,407),(43,113,408),(44,114,341),(45,115,342),(46,116,343),(47,117,344),(48,118,345),(49,119,346),(50,120,347),(51,121,348),(52,122,349),(53,123,350),(54,124,351),(55,125,352),(56,126,353),(57,127,354),(58,128,355),(59,129,356),(60,130,357),(61,131,358),(62,132,359),(63,133,360),(64,134,361),(65,135,362),(66,136,363),(67,69,364),(68,70,365),(137,262,293),(138,263,294),(139,264,295),(140,265,296),(141,266,297),(142,267,298),(143,268,299),(144,269,300),(145,270,301),(146,271,302),(147,272,303),(148,205,304),(149,206,305),(150,207,306),(151,208,307),(152,209,308),(153,210,309),(154,211,310),(155,212,311),(156,213,312),(157,214,313),(158,215,314),(159,216,315),(160,217,316),(161,218,317),(162,219,318),(163,220,319),(164,221,320),(165,222,321),(166,223,322),(167,224,323),(168,225,324),(169,226,325),(170,227,326),(171,228,327),(172,229,328),(173,230,329),(174,231,330),(175,232,331),(176,233,332),(177,234,333),(178,235,334),(179,236,335),(180,237,336),(181,238,337),(182,239,338),(183,240,339),(184,241,340),(185,242,273),(186,243,274),(187,244,275),(188,245,276),(189,246,277),(190,247,278),(191,248,279),(192,249,280),(193,250,281),(194,251,282),(195,252,283),(196,253,284),(197,254,285),(198,255,286),(199,256,287),(200,257,288),(201,258,289),(202,259,290),(203,260,291),(204,261,292)], [(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),(341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408)], [(1,245,35,211),(2,244,36,210),(3,243,37,209),(4,242,38,208),(5,241,39,207),(6,240,40,206),(7,239,41,205),(8,238,42,272),(9,237,43,271),(10,236,44,270),(11,235,45,269),(12,234,46,268),(13,233,47,267),(14,232,48,266),(15,231,49,265),(16,230,50,264),(17,229,51,263),(18,228,52,262),(19,227,53,261),(20,226,54,260),(21,225,55,259),(22,224,56,258),(23,223,57,257),(24,222,58,256),(25,221,59,255),(26,220,60,254),(27,219,61,253),(28,218,62,252),(29,217,63,251),(30,216,64,250),(31,215,65,249),(32,214,66,248),(33,213,67,247),(34,212,68,246),(69,278,103,312),(70,277,104,311),(71,276,105,310),(72,275,106,309),(73,274,107,308),(74,273,108,307),(75,340,109,306),(76,339,110,305),(77,338,111,304),(78,337,112,303),(79,336,113,302),(80,335,114,301),(81,334,115,300),(82,333,116,299),(83,332,117,298),(84,331,118,297),(85,330,119,296),(86,329,120,295),(87,328,121,294),(88,327,122,293),(89,326,123,292),(90,325,124,291),(91,324,125,290),(92,323,126,289),(93,322,127,288),(94,321,128,287),(95,320,129,286),(96,319,130,285),(97,318,131,284),(98,317,132,283),(99,316,133,282),(100,315,134,281),(101,314,135,280),(102,313,136,279),(137,383,171,349),(138,382,172,348),(139,381,173,347),(140,380,174,346),(141,379,175,345),(142,378,176,344),(143,377,177,343),(144,376,178,342),(145,375,179,341),(146,374,180,408),(147,373,181,407),(148,372,182,406),(149,371,183,405),(150,370,184,404),(151,369,185,403),(152,368,186,402),(153,367,187,401),(154,366,188,400),(155,365,189,399),(156,364,190,398),(157,363,191,397),(158,362,192,396),(159,361,193,395),(160,360,194,394),(161,359,195,393),(162,358,196,392),(163,357,197,391),(164,356,198,390),(165,355,199,389),(166,354,200,388),(167,353,201,387),(168,352,202,386),(169,351,203,385),(170,350,204,384)]])
111 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 12A | 12B | 12C | 12D | 12E | 12F | 17A | ··· | 17H | 34A | ··· | 34H | 51A | ··· | 51P | 68A | ··· | 68P | 102A | ··· | 102P | 204A | ··· | 204AF |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 68 | ··· | 68 | 102 | ··· | 102 | 204 | ··· | 204 |
| size | 1 | 1 | 1 | 1 | 2 | 34 | 34 | 1 | 1 | 2 | 2 | 34 | 34 | 34 | 34 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
111 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | Q8 | C3×Q8 | D17 | D34 | C3×D17 | Dic34 | C6×D17 | C3×Dic34 |
| kernel | C3×Dic34 | C3×Dic17 | C204 | Dic34 | Dic17 | C68 | C51 | C17 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 2 | 8 | 8 | 16 | 16 | 16 | 32 |
Matrix representation of C3×Dic34 ►in GL4(𝔽409) generated by
| 355 | 0 | 0 | 0 |
| 0 | 355 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 408 | 205 | 0 | 0 |
| 0 | 0 | 1 | 206 |
| 0 | 0 | 135 | 408 |
| 359 | 220 | 0 | 0 |
| 195 | 50 | 0 | 0 |
| 0 | 0 | 227 | 194 |
| 0 | 0 | 97 | 182 |
G:=sub<GL(4,GF(409))| [355,0,0,0,0,355,0,0,0,0,1,0,0,0,0,1],[0,408,0,0,1,205,0,0,0,0,1,135,0,0,206,408],[359,195,0,0,220,50,0,0,0,0,227,97,0,0,194,182] >;
C3×Dic34 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_{34} % in TeX
G:=Group("C3xDic34"); // GroupNames label
G:=SmallGroup(408,15);
// by ID
G=gap.SmallGroup(408,15);
# by ID
G:=PCGroup([5,-2,-2,-3,-2,-17,60,141,66,9604]);
// Polycyclic
G:=Group<a,b,c|a^3=b^68=1,c^2=b^34,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export