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G = C3×Dic34order 408 = 23·3·17

Direct product of C3 and Dic34

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C3×Dic34, C513Q8, 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

C1C34 — C3×Dic34
C1C17C34C102C3×Dic17 — C3×Dic34
C17C34 — C3×Dic34
C1C6C12

Generators and relations for C3×Dic34
 G = < a,b,c | a3=b68=1, c2=b34, ab=ba, ac=ca, cbc-1=b-1 >

17C4
17C4
17Q8
17C12
17C12
17C3×Q8

Smallest permutation representation of C3×Dic34
Regular action on 408 points
Generators in S408
(1 89 330)(2 90 331)(3 91 332)(4 92 333)(5 93 334)(6 94 335)(7 95 336)(8 96 337)(9 97 338)(10 98 339)(11 99 340)(12 100 273)(13 101 274)(14 102 275)(15 103 276)(16 104 277)(17 105 278)(18 106 279)(19 107 280)(20 108 281)(21 109 282)(22 110 283)(23 111 284)(24 112 285)(25 113 286)(26 114 287)(27 115 288)(28 116 289)(29 117 290)(30 118 291)(31 119 292)(32 120 293)(33 121 294)(34 122 295)(35 123 296)(36 124 297)(37 125 298)(38 126 299)(39 127 300)(40 128 301)(41 129 302)(42 130 303)(43 131 304)(44 132 305)(45 133 306)(46 134 307)(47 135 308)(48 136 309)(49 69 310)(50 70 311)(51 71 312)(52 72 313)(53 73 314)(54 74 315)(55 75 316)(56 76 317)(57 77 318)(58 78 319)(59 79 320)(60 80 321)(61 81 322)(62 82 323)(63 83 324)(64 84 325)(65 85 326)(66 86 327)(67 87 328)(68 88 329)(137 365 247)(138 366 248)(139 367 249)(140 368 250)(141 369 251)(142 370 252)(143 371 253)(144 372 254)(145 373 255)(146 374 256)(147 375 257)(148 376 258)(149 377 259)(150 378 260)(151 379 261)(152 380 262)(153 381 263)(154 382 264)(155 383 265)(156 384 266)(157 385 267)(158 386 268)(159 387 269)(160 388 270)(161 389 271)(162 390 272)(163 391 205)(164 392 206)(165 393 207)(166 394 208)(167 395 209)(168 396 210)(169 397 211)(170 398 212)(171 399 213)(172 400 214)(173 401 215)(174 402 216)(175 403 217)(176 404 218)(177 405 219)(178 406 220)(179 407 221)(180 408 222)(181 341 223)(182 342 224)(183 343 225)(184 344 226)(185 345 227)(186 346 228)(187 347 229)(188 348 230)(189 349 231)(190 350 232)(191 351 233)(192 352 234)(193 353 235)(194 354 236)(195 355 237)(196 356 238)(197 357 239)(198 358 240)(199 359 241)(200 360 242)(201 361 243)(202 362 244)(203 363 245)(204 364 246)
(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)
(1 403 35 369)(2 402 36 368)(3 401 37 367)(4 400 38 366)(5 399 39 365)(6 398 40 364)(7 397 41 363)(8 396 42 362)(9 395 43 361)(10 394 44 360)(11 393 45 359)(12 392 46 358)(13 391 47 357)(14 390 48 356)(15 389 49 355)(16 388 50 354)(17 387 51 353)(18 386 52 352)(19 385 53 351)(20 384 54 350)(21 383 55 349)(22 382 56 348)(23 381 57 347)(24 380 58 346)(25 379 59 345)(26 378 60 344)(27 377 61 343)(28 376 62 342)(29 375 63 341)(30 374 64 408)(31 373 65 407)(32 372 66 406)(33 371 67 405)(34 370 68 404)(69 237 103 271)(70 236 104 270)(71 235 105 269)(72 234 106 268)(73 233 107 267)(74 232 108 266)(75 231 109 265)(76 230 110 264)(77 229 111 263)(78 228 112 262)(79 227 113 261)(80 226 114 260)(81 225 115 259)(82 224 116 258)(83 223 117 257)(84 222 118 256)(85 221 119 255)(86 220 120 254)(87 219 121 253)(88 218 122 252)(89 217 123 251)(90 216 124 250)(91 215 125 249)(92 214 126 248)(93 213 127 247)(94 212 128 246)(95 211 129 245)(96 210 130 244)(97 209 131 243)(98 208 132 242)(99 207 133 241)(100 206 134 240)(101 205 135 239)(102 272 136 238)(137 334 171 300)(138 333 172 299)(139 332 173 298)(140 331 174 297)(141 330 175 296)(142 329 176 295)(143 328 177 294)(144 327 178 293)(145 326 179 292)(146 325 180 291)(147 324 181 290)(148 323 182 289)(149 322 183 288)(150 321 184 287)(151 320 185 286)(152 319 186 285)(153 318 187 284)(154 317 188 283)(155 316 189 282)(156 315 190 281)(157 314 191 280)(158 313 192 279)(159 312 193 278)(160 311 194 277)(161 310 195 276)(162 309 196 275)(163 308 197 274)(164 307 198 273)(165 306 199 340)(166 305 200 339)(167 304 201 338)(168 303 202 337)(169 302 203 336)(170 301 204 335)

G:=sub<Sym(408)| (1,89,330)(2,90,331)(3,91,332)(4,92,333)(5,93,334)(6,94,335)(7,95,336)(8,96,337)(9,97,338)(10,98,339)(11,99,340)(12,100,273)(13,101,274)(14,102,275)(15,103,276)(16,104,277)(17,105,278)(18,106,279)(19,107,280)(20,108,281)(21,109,282)(22,110,283)(23,111,284)(24,112,285)(25,113,286)(26,114,287)(27,115,288)(28,116,289)(29,117,290)(30,118,291)(31,119,292)(32,120,293)(33,121,294)(34,122,295)(35,123,296)(36,124,297)(37,125,298)(38,126,299)(39,127,300)(40,128,301)(41,129,302)(42,130,303)(43,131,304)(44,132,305)(45,133,306)(46,134,307)(47,135,308)(48,136,309)(49,69,310)(50,70,311)(51,71,312)(52,72,313)(53,73,314)(54,74,315)(55,75,316)(56,76,317)(57,77,318)(58,78,319)(59,79,320)(60,80,321)(61,81,322)(62,82,323)(63,83,324)(64,84,325)(65,85,326)(66,86,327)(67,87,328)(68,88,329)(137,365,247)(138,366,248)(139,367,249)(140,368,250)(141,369,251)(142,370,252)(143,371,253)(144,372,254)(145,373,255)(146,374,256)(147,375,257)(148,376,258)(149,377,259)(150,378,260)(151,379,261)(152,380,262)(153,381,263)(154,382,264)(155,383,265)(156,384,266)(157,385,267)(158,386,268)(159,387,269)(160,388,270)(161,389,271)(162,390,272)(163,391,205)(164,392,206)(165,393,207)(166,394,208)(167,395,209)(168,396,210)(169,397,211)(170,398,212)(171,399,213)(172,400,214)(173,401,215)(174,402,216)(175,403,217)(176,404,218)(177,405,219)(178,406,220)(179,407,221)(180,408,222)(181,341,223)(182,342,224)(183,343,225)(184,344,226)(185,345,227)(186,346,228)(187,347,229)(188,348,230)(189,349,231)(190,350,232)(191,351,233)(192,352,234)(193,353,235)(194,354,236)(195,355,237)(196,356,238)(197,357,239)(198,358,240)(199,359,241)(200,360,242)(201,361,243)(202,362,244)(203,363,245)(204,364,246), (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), (1,403,35,369)(2,402,36,368)(3,401,37,367)(4,400,38,366)(5,399,39,365)(6,398,40,364)(7,397,41,363)(8,396,42,362)(9,395,43,361)(10,394,44,360)(11,393,45,359)(12,392,46,358)(13,391,47,357)(14,390,48,356)(15,389,49,355)(16,388,50,354)(17,387,51,353)(18,386,52,352)(19,385,53,351)(20,384,54,350)(21,383,55,349)(22,382,56,348)(23,381,57,347)(24,380,58,346)(25,379,59,345)(26,378,60,344)(27,377,61,343)(28,376,62,342)(29,375,63,341)(30,374,64,408)(31,373,65,407)(32,372,66,406)(33,371,67,405)(34,370,68,404)(69,237,103,271)(70,236,104,270)(71,235,105,269)(72,234,106,268)(73,233,107,267)(74,232,108,266)(75,231,109,265)(76,230,110,264)(77,229,111,263)(78,228,112,262)(79,227,113,261)(80,226,114,260)(81,225,115,259)(82,224,116,258)(83,223,117,257)(84,222,118,256)(85,221,119,255)(86,220,120,254)(87,219,121,253)(88,218,122,252)(89,217,123,251)(90,216,124,250)(91,215,125,249)(92,214,126,248)(93,213,127,247)(94,212,128,246)(95,211,129,245)(96,210,130,244)(97,209,131,243)(98,208,132,242)(99,207,133,241)(100,206,134,240)(101,205,135,239)(102,272,136,238)(137,334,171,300)(138,333,172,299)(139,332,173,298)(140,331,174,297)(141,330,175,296)(142,329,176,295)(143,328,177,294)(144,327,178,293)(145,326,179,292)(146,325,180,291)(147,324,181,290)(148,323,182,289)(149,322,183,288)(150,321,184,287)(151,320,185,286)(152,319,186,285)(153,318,187,284)(154,317,188,283)(155,316,189,282)(156,315,190,281)(157,314,191,280)(158,313,192,279)(159,312,193,278)(160,311,194,277)(161,310,195,276)(162,309,196,275)(163,308,197,274)(164,307,198,273)(165,306,199,340)(166,305,200,339)(167,304,201,338)(168,303,202,337)(169,302,203,336)(170,301,204,335)>;

G:=Group( (1,89,330)(2,90,331)(3,91,332)(4,92,333)(5,93,334)(6,94,335)(7,95,336)(8,96,337)(9,97,338)(10,98,339)(11,99,340)(12,100,273)(13,101,274)(14,102,275)(15,103,276)(16,104,277)(17,105,278)(18,106,279)(19,107,280)(20,108,281)(21,109,282)(22,110,283)(23,111,284)(24,112,285)(25,113,286)(26,114,287)(27,115,288)(28,116,289)(29,117,290)(30,118,291)(31,119,292)(32,120,293)(33,121,294)(34,122,295)(35,123,296)(36,124,297)(37,125,298)(38,126,299)(39,127,300)(40,128,301)(41,129,302)(42,130,303)(43,131,304)(44,132,305)(45,133,306)(46,134,307)(47,135,308)(48,136,309)(49,69,310)(50,70,311)(51,71,312)(52,72,313)(53,73,314)(54,74,315)(55,75,316)(56,76,317)(57,77,318)(58,78,319)(59,79,320)(60,80,321)(61,81,322)(62,82,323)(63,83,324)(64,84,325)(65,85,326)(66,86,327)(67,87,328)(68,88,329)(137,365,247)(138,366,248)(139,367,249)(140,368,250)(141,369,251)(142,370,252)(143,371,253)(144,372,254)(145,373,255)(146,374,256)(147,375,257)(148,376,258)(149,377,259)(150,378,260)(151,379,261)(152,380,262)(153,381,263)(154,382,264)(155,383,265)(156,384,266)(157,385,267)(158,386,268)(159,387,269)(160,388,270)(161,389,271)(162,390,272)(163,391,205)(164,392,206)(165,393,207)(166,394,208)(167,395,209)(168,396,210)(169,397,211)(170,398,212)(171,399,213)(172,400,214)(173,401,215)(174,402,216)(175,403,217)(176,404,218)(177,405,219)(178,406,220)(179,407,221)(180,408,222)(181,341,223)(182,342,224)(183,343,225)(184,344,226)(185,345,227)(186,346,228)(187,347,229)(188,348,230)(189,349,231)(190,350,232)(191,351,233)(192,352,234)(193,353,235)(194,354,236)(195,355,237)(196,356,238)(197,357,239)(198,358,240)(199,359,241)(200,360,242)(201,361,243)(202,362,244)(203,363,245)(204,364,246), (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), (1,403,35,369)(2,402,36,368)(3,401,37,367)(4,400,38,366)(5,399,39,365)(6,398,40,364)(7,397,41,363)(8,396,42,362)(9,395,43,361)(10,394,44,360)(11,393,45,359)(12,392,46,358)(13,391,47,357)(14,390,48,356)(15,389,49,355)(16,388,50,354)(17,387,51,353)(18,386,52,352)(19,385,53,351)(20,384,54,350)(21,383,55,349)(22,382,56,348)(23,381,57,347)(24,380,58,346)(25,379,59,345)(26,378,60,344)(27,377,61,343)(28,376,62,342)(29,375,63,341)(30,374,64,408)(31,373,65,407)(32,372,66,406)(33,371,67,405)(34,370,68,404)(69,237,103,271)(70,236,104,270)(71,235,105,269)(72,234,106,268)(73,233,107,267)(74,232,108,266)(75,231,109,265)(76,230,110,264)(77,229,111,263)(78,228,112,262)(79,227,113,261)(80,226,114,260)(81,225,115,259)(82,224,116,258)(83,223,117,257)(84,222,118,256)(85,221,119,255)(86,220,120,254)(87,219,121,253)(88,218,122,252)(89,217,123,251)(90,216,124,250)(91,215,125,249)(92,214,126,248)(93,213,127,247)(94,212,128,246)(95,211,129,245)(96,210,130,244)(97,209,131,243)(98,208,132,242)(99,207,133,241)(100,206,134,240)(101,205,135,239)(102,272,136,238)(137,334,171,300)(138,333,172,299)(139,332,173,298)(140,331,174,297)(141,330,175,296)(142,329,176,295)(143,328,177,294)(144,327,178,293)(145,326,179,292)(146,325,180,291)(147,324,181,290)(148,323,182,289)(149,322,183,288)(150,321,184,287)(151,320,185,286)(152,319,186,285)(153,318,187,284)(154,317,188,283)(155,316,189,282)(156,315,190,281)(157,314,191,280)(158,313,192,279)(159,312,193,278)(160,311,194,277)(161,310,195,276)(162,309,196,275)(163,308,197,274)(164,307,198,273)(165,306,199,340)(166,305,200,339)(167,304,201,338)(168,303,202,337)(169,302,203,336)(170,301,204,335) );

G=PermutationGroup([(1,89,330),(2,90,331),(3,91,332),(4,92,333),(5,93,334),(6,94,335),(7,95,336),(8,96,337),(9,97,338),(10,98,339),(11,99,340),(12,100,273),(13,101,274),(14,102,275),(15,103,276),(16,104,277),(17,105,278),(18,106,279),(19,107,280),(20,108,281),(21,109,282),(22,110,283),(23,111,284),(24,112,285),(25,113,286),(26,114,287),(27,115,288),(28,116,289),(29,117,290),(30,118,291),(31,119,292),(32,120,293),(33,121,294),(34,122,295),(35,123,296),(36,124,297),(37,125,298),(38,126,299),(39,127,300),(40,128,301),(41,129,302),(42,130,303),(43,131,304),(44,132,305),(45,133,306),(46,134,307),(47,135,308),(48,136,309),(49,69,310),(50,70,311),(51,71,312),(52,72,313),(53,73,314),(54,74,315),(55,75,316),(56,76,317),(57,77,318),(58,78,319),(59,79,320),(60,80,321),(61,81,322),(62,82,323),(63,83,324),(64,84,325),(65,85,326),(66,86,327),(67,87,328),(68,88,329),(137,365,247),(138,366,248),(139,367,249),(140,368,250),(141,369,251),(142,370,252),(143,371,253),(144,372,254),(145,373,255),(146,374,256),(147,375,257),(148,376,258),(149,377,259),(150,378,260),(151,379,261),(152,380,262),(153,381,263),(154,382,264),(155,383,265),(156,384,266),(157,385,267),(158,386,268),(159,387,269),(160,388,270),(161,389,271),(162,390,272),(163,391,205),(164,392,206),(165,393,207),(166,394,208),(167,395,209),(168,396,210),(169,397,211),(170,398,212),(171,399,213),(172,400,214),(173,401,215),(174,402,216),(175,403,217),(176,404,218),(177,405,219),(178,406,220),(179,407,221),(180,408,222),(181,341,223),(182,342,224),(183,343,225),(184,344,226),(185,345,227),(186,346,228),(187,347,229),(188,348,230),(189,349,231),(190,350,232),(191,351,233),(192,352,234),(193,353,235),(194,354,236),(195,355,237),(196,356,238),(197,357,239),(198,358,240),(199,359,241),(200,360,242),(201,361,243),(202,362,244),(203,363,245),(204,364,246)], [(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)], [(1,403,35,369),(2,402,36,368),(3,401,37,367),(4,400,38,366),(5,399,39,365),(6,398,40,364),(7,397,41,363),(8,396,42,362),(9,395,43,361),(10,394,44,360),(11,393,45,359),(12,392,46,358),(13,391,47,357),(14,390,48,356),(15,389,49,355),(16,388,50,354),(17,387,51,353),(18,386,52,352),(19,385,53,351),(20,384,54,350),(21,383,55,349),(22,382,56,348),(23,381,57,347),(24,380,58,346),(25,379,59,345),(26,378,60,344),(27,377,61,343),(28,376,62,342),(29,375,63,341),(30,374,64,408),(31,373,65,407),(32,372,66,406),(33,371,67,405),(34,370,68,404),(69,237,103,271),(70,236,104,270),(71,235,105,269),(72,234,106,268),(73,233,107,267),(74,232,108,266),(75,231,109,265),(76,230,110,264),(77,229,111,263),(78,228,112,262),(79,227,113,261),(80,226,114,260),(81,225,115,259),(82,224,116,258),(83,223,117,257),(84,222,118,256),(85,221,119,255),(86,220,120,254),(87,219,121,253),(88,218,122,252),(89,217,123,251),(90,216,124,250),(91,215,125,249),(92,214,126,248),(93,213,127,247),(94,212,128,246),(95,211,129,245),(96,210,130,244),(97,209,131,243),(98,208,132,242),(99,207,133,241),(100,206,134,240),(101,205,135,239),(102,272,136,238),(137,334,171,300),(138,333,172,299),(139,332,173,298),(140,331,174,297),(141,330,175,296),(142,329,176,295),(143,328,177,294),(144,327,178,293),(145,326,179,292),(146,325,180,291),(147,324,181,290),(148,323,182,289),(149,322,183,288),(150,321,184,287),(151,320,185,286),(152,319,186,285),(153,318,187,284),(154,317,188,283),(155,316,189,282),(156,315,190,281),(157,314,191,280),(158,313,192,279),(159,312,193,278),(160,311,194,277),(161,310,195,276),(162,309,196,275),(163,308,197,274),(164,307,198,273),(165,306,199,340),(166,305,200,339),(167,304,201,338),(168,303,202,337),(169,302,203,336),(170,301,204,335)])

111 conjugacy classes

class 1  2 3A3B4A4B4C6A6B12A12B12C12D12E12F17A···17H34A···34H51A···51P68A···68P102A···102P204A···204AF
order12334446612121212121217···1734···3451···5168···68102···102204···204
size1111234341122343434342···22···22···22···22···22···2

111 irreducible representations

dim11111122222222
type+++-++-
imageC1C2C2C3C6C6Q8C3×Q8D17D34C3×D17Dic34C6×D17C3×Dic34
kernelC3×Dic34C3×Dic17C204Dic34Dic17C68C51C17C12C6C4C3C2C1
# reps121242128816161632

Matrix representation of C3×Dic34 in GL4(𝔽409) generated by

355000
035500
0010
0001
,
0100
40820500
001206
00135408
,
35922000
1955000
00227194
0097182
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

Subgroup lattice of C3×Dic34 in TeX

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