direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: Dic3×C34, C6⋊C68, C102⋊5C4, C34.16D6, C102.21C22, C3⋊2(C2×C68), (C2×C6).C34, C51⋊12(C2×C4), C2.2(S3×C34), (C2×C34).2S3, C6.4(C2×C34), C22.(S3×C17), (C2×C102).3C2, SmallGroup(408,23)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C34 |
Generators and relations for Dic3×C34
G = < a,b,c | a34=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Dic3×C34
►Regular action on 408 points
Generators in S408
(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 66 333 261 181 358)(2 67 334 262 182 359)(3 68 335 263 183 360)(4 35 336 264 184 361)(5 36 337 265 185 362)(6 37 338 266 186 363)(7 38 339 267 187 364)(8 39 340 268 188 365)(9 40 307 269 189 366)(10 41 308 270 190 367)(11 42 309 271 191 368)(12 43 310 272 192 369)(13 44 311 239 193 370)(14 45 312 240 194 371)(15 46 313 241 195 372)(16 47 314 242 196 373)(17 48 315 243 197 374)(18 49 316 244 198 341)(19 50 317 245 199 342)(20 51 318 246 200 343)(21 52 319 247 201 344)(22 53 320 248 202 345)(23 54 321 249 203 346)(24 55 322 250 204 347)(25 56 323 251 171 348)(26 57 324 252 172 349)(27 58 325 253 173 350)(28 59 326 254 174 351)(29 60 327 255 175 352)(30 61 328 256 176 353)(31 62 329 257 177 354)(32 63 330 258 178 355)(33 64 331 259 179 356)(34 65 332 260 180 357)(69 217 123 382 301 160)(70 218 124 383 302 161)(71 219 125 384 303 162)(72 220 126 385 304 163)(73 221 127 386 305 164)(74 222 128 387 306 165)(75 223 129 388 273 166)(76 224 130 389 274 167)(77 225 131 390 275 168)(78 226 132 391 276 169)(79 227 133 392 277 170)(80 228 134 393 278 137)(81 229 135 394 279 138)(82 230 136 395 280 139)(83 231 103 396 281 140)(84 232 104 397 282 141)(85 233 105 398 283 142)(86 234 106 399 284 143)(87 235 107 400 285 144)(88 236 108 401 286 145)(89 237 109 402 287 146)(90 238 110 403 288 147)(91 205 111 404 289 148)(92 206 112 405 290 149)(93 207 113 406 291 150)(94 208 114 407 292 151)(95 209 115 408 293 152)(96 210 116 375 294 153)(97 211 117 376 295 154)(98 212 118 377 296 155)(99 213 119 378 297 156)(100 214 120 379 298 157)(101 215 121 380 299 158)(102 216 122 381 300 159)
(1 221 261 305)(2 222 262 306)(3 223 263 273)(4 224 264 274)(5 225 265 275)(6 226 266 276)(7 227 267 277)(8 228 268 278)(9 229 269 279)(10 230 270 280)(11 231 271 281)(12 232 272 282)(13 233 239 283)(14 234 240 284)(15 235 241 285)(16 236 242 286)(17 237 243 287)(18 238 244 288)(19 205 245 289)(20 206 246 290)(21 207 247 291)(22 208 248 292)(23 209 249 293)(24 210 250 294)(25 211 251 295)(26 212 252 296)(27 213 253 297)(28 214 254 298)(29 215 255 299)(30 216 256 300)(31 217 257 301)(32 218 258 302)(33 219 259 303)(34 220 260 304)(35 76 184 389)(36 77 185 390)(37 78 186 391)(38 79 187 392)(39 80 188 393)(40 81 189 394)(41 82 190 395)(42 83 191 396)(43 84 192 397)(44 85 193 398)(45 86 194 399)(46 87 195 400)(47 88 196 401)(48 89 197 402)(49 90 198 403)(50 91 199 404)(51 92 200 405)(52 93 201 406)(53 94 202 407)(54 95 203 408)(55 96 204 375)(56 97 171 376)(57 98 172 377)(58 99 173 378)(59 100 174 379)(60 101 175 380)(61 102 176 381)(62 69 177 382)(63 70 178 383)(64 71 179 384)(65 72 180 385)(66 73 181 386)(67 74 182 387)(68 75 183 388)(103 309 140 368)(104 310 141 369)(105 311 142 370)(106 312 143 371)(107 313 144 372)(108 314 145 373)(109 315 146 374)(110 316 147 341)(111 317 148 342)(112 318 149 343)(113 319 150 344)(114 320 151 345)(115 321 152 346)(116 322 153 347)(117 323 154 348)(118 324 155 349)(119 325 156 350)(120 326 157 351)(121 327 158 352)(122 328 159 353)(123 329 160 354)(124 330 161 355)(125 331 162 356)(126 332 163 357)(127 333 164 358)(128 334 165 359)(129 335 166 360)(130 336 167 361)(131 337 168 362)(132 338 169 363)(133 339 170 364)(134 340 137 365)(135 307 138 366)(136 308 139 367)
G:=sub<Sym(408)| (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), (1,66,333,261,181,358)(2,67,334,262,182,359)(3,68,335,263,183,360)(4,35,336,264,184,361)(5,36,337,265,185,362)(6,37,338,266,186,363)(7,38,339,267,187,364)(8,39,340,268,188,365)(9,40,307,269,189,366)(10,41,308,270,190,367)(11,42,309,271,191,368)(12,43,310,272,192,369)(13,44,311,239,193,370)(14,45,312,240,194,371)(15,46,313,241,195,372)(16,47,314,242,196,373)(17,48,315,243,197,374)(18,49,316,244,198,341)(19,50,317,245,199,342)(20,51,318,246,200,343)(21,52,319,247,201,344)(22,53,320,248,202,345)(23,54,321,249,203,346)(24,55,322,250,204,347)(25,56,323,251,171,348)(26,57,324,252,172,349)(27,58,325,253,173,350)(28,59,326,254,174,351)(29,60,327,255,175,352)(30,61,328,256,176,353)(31,62,329,257,177,354)(32,63,330,258,178,355)(33,64,331,259,179,356)(34,65,332,260,180,357)(69,217,123,382,301,160)(70,218,124,383,302,161)(71,219,125,384,303,162)(72,220,126,385,304,163)(73,221,127,386,305,164)(74,222,128,387,306,165)(75,223,129,388,273,166)(76,224,130,389,274,167)(77,225,131,390,275,168)(78,226,132,391,276,169)(79,227,133,392,277,170)(80,228,134,393,278,137)(81,229,135,394,279,138)(82,230,136,395,280,139)(83,231,103,396,281,140)(84,232,104,397,282,141)(85,233,105,398,283,142)(86,234,106,399,284,143)(87,235,107,400,285,144)(88,236,108,401,286,145)(89,237,109,402,287,146)(90,238,110,403,288,147)(91,205,111,404,289,148)(92,206,112,405,290,149)(93,207,113,406,291,150)(94,208,114,407,292,151)(95,209,115,408,293,152)(96,210,116,375,294,153)(97,211,117,376,295,154)(98,212,118,377,296,155)(99,213,119,378,297,156)(100,214,120,379,298,157)(101,215,121,380,299,158)(102,216,122,381,300,159), (1,221,261,305)(2,222,262,306)(3,223,263,273)(4,224,264,274)(5,225,265,275)(6,226,266,276)(7,227,267,277)(8,228,268,278)(9,229,269,279)(10,230,270,280)(11,231,271,281)(12,232,272,282)(13,233,239,283)(14,234,240,284)(15,235,241,285)(16,236,242,286)(17,237,243,287)(18,238,244,288)(19,205,245,289)(20,206,246,290)(21,207,247,291)(22,208,248,292)(23,209,249,293)(24,210,250,294)(25,211,251,295)(26,212,252,296)(27,213,253,297)(28,214,254,298)(29,215,255,299)(30,216,256,300)(31,217,257,301)(32,218,258,302)(33,219,259,303)(34,220,260,304)(35,76,184,389)(36,77,185,390)(37,78,186,391)(38,79,187,392)(39,80,188,393)(40,81,189,394)(41,82,190,395)(42,83,191,396)(43,84,192,397)(44,85,193,398)(45,86,194,399)(46,87,195,400)(47,88,196,401)(48,89,197,402)(49,90,198,403)(50,91,199,404)(51,92,200,405)(52,93,201,406)(53,94,202,407)(54,95,203,408)(55,96,204,375)(56,97,171,376)(57,98,172,377)(58,99,173,378)(59,100,174,379)(60,101,175,380)(61,102,176,381)(62,69,177,382)(63,70,178,383)(64,71,179,384)(65,72,180,385)(66,73,181,386)(67,74,182,387)(68,75,183,388)(103,309,140,368)(104,310,141,369)(105,311,142,370)(106,312,143,371)(107,313,144,372)(108,314,145,373)(109,315,146,374)(110,316,147,341)(111,317,148,342)(112,318,149,343)(113,319,150,344)(114,320,151,345)(115,321,152,346)(116,322,153,347)(117,323,154,348)(118,324,155,349)(119,325,156,350)(120,326,157,351)(121,327,158,352)(122,328,159,353)(123,329,160,354)(124,330,161,355)(125,331,162,356)(126,332,163,357)(127,333,164,358)(128,334,165,359)(129,335,166,360)(130,336,167,361)(131,337,168,362)(132,338,169,363)(133,339,170,364)(134,340,137,365)(135,307,138,366)(136,308,139,367)>;
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)(239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272)(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), (1,66,333,261,181,358)(2,67,334,262,182,359)(3,68,335,263,183,360)(4,35,336,264,184,361)(5,36,337,265,185,362)(6,37,338,266,186,363)(7,38,339,267,187,364)(8,39,340,268,188,365)(9,40,307,269,189,366)(10,41,308,270,190,367)(11,42,309,271,191,368)(12,43,310,272,192,369)(13,44,311,239,193,370)(14,45,312,240,194,371)(15,46,313,241,195,372)(16,47,314,242,196,373)(17,48,315,243,197,374)(18,49,316,244,198,341)(19,50,317,245,199,342)(20,51,318,246,200,343)(21,52,319,247,201,344)(22,53,320,248,202,345)(23,54,321,249,203,346)(24,55,322,250,204,347)(25,56,323,251,171,348)(26,57,324,252,172,349)(27,58,325,253,173,350)(28,59,326,254,174,351)(29,60,327,255,175,352)(30,61,328,256,176,353)(31,62,329,257,177,354)(32,63,330,258,178,355)(33,64,331,259,179,356)(34,65,332,260,180,357)(69,217,123,382,301,160)(70,218,124,383,302,161)(71,219,125,384,303,162)(72,220,126,385,304,163)(73,221,127,386,305,164)(74,222,128,387,306,165)(75,223,129,388,273,166)(76,224,130,389,274,167)(77,225,131,390,275,168)(78,226,132,391,276,169)(79,227,133,392,277,170)(80,228,134,393,278,137)(81,229,135,394,279,138)(82,230,136,395,280,139)(83,231,103,396,281,140)(84,232,104,397,282,141)(85,233,105,398,283,142)(86,234,106,399,284,143)(87,235,107,400,285,144)(88,236,108,401,286,145)(89,237,109,402,287,146)(90,238,110,403,288,147)(91,205,111,404,289,148)(92,206,112,405,290,149)(93,207,113,406,291,150)(94,208,114,407,292,151)(95,209,115,408,293,152)(96,210,116,375,294,153)(97,211,117,376,295,154)(98,212,118,377,296,155)(99,213,119,378,297,156)(100,214,120,379,298,157)(101,215,121,380,299,158)(102,216,122,381,300,159), (1,221,261,305)(2,222,262,306)(3,223,263,273)(4,224,264,274)(5,225,265,275)(6,226,266,276)(7,227,267,277)(8,228,268,278)(9,229,269,279)(10,230,270,280)(11,231,271,281)(12,232,272,282)(13,233,239,283)(14,234,240,284)(15,235,241,285)(16,236,242,286)(17,237,243,287)(18,238,244,288)(19,205,245,289)(20,206,246,290)(21,207,247,291)(22,208,248,292)(23,209,249,293)(24,210,250,294)(25,211,251,295)(26,212,252,296)(27,213,253,297)(28,214,254,298)(29,215,255,299)(30,216,256,300)(31,217,257,301)(32,218,258,302)(33,219,259,303)(34,220,260,304)(35,76,184,389)(36,77,185,390)(37,78,186,391)(38,79,187,392)(39,80,188,393)(40,81,189,394)(41,82,190,395)(42,83,191,396)(43,84,192,397)(44,85,193,398)(45,86,194,399)(46,87,195,400)(47,88,196,401)(48,89,197,402)(49,90,198,403)(50,91,199,404)(51,92,200,405)(52,93,201,406)(53,94,202,407)(54,95,203,408)(55,96,204,375)(56,97,171,376)(57,98,172,377)(58,99,173,378)(59,100,174,379)(60,101,175,380)(61,102,176,381)(62,69,177,382)(63,70,178,383)(64,71,179,384)(65,72,180,385)(66,73,181,386)(67,74,182,387)(68,75,183,388)(103,309,140,368)(104,310,141,369)(105,311,142,370)(106,312,143,371)(107,313,144,372)(108,314,145,373)(109,315,146,374)(110,316,147,341)(111,317,148,342)(112,318,149,343)(113,319,150,344)(114,320,151,345)(115,321,152,346)(116,322,153,347)(117,323,154,348)(118,324,155,349)(119,325,156,350)(120,326,157,351)(121,327,158,352)(122,328,159,353)(123,329,160,354)(124,330,161,355)(125,331,162,356)(126,332,163,357)(127,333,164,358)(128,334,165,359)(129,335,166,360)(130,336,167,361)(131,337,168,362)(132,338,169,363)(133,339,170,364)(134,340,137,365)(135,307,138,366)(136,308,139,367) );
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),(239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272),(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)], [(1,66,333,261,181,358),(2,67,334,262,182,359),(3,68,335,263,183,360),(4,35,336,264,184,361),(5,36,337,265,185,362),(6,37,338,266,186,363),(7,38,339,267,187,364),(8,39,340,268,188,365),(9,40,307,269,189,366),(10,41,308,270,190,367),(11,42,309,271,191,368),(12,43,310,272,192,369),(13,44,311,239,193,370),(14,45,312,240,194,371),(15,46,313,241,195,372),(16,47,314,242,196,373),(17,48,315,243,197,374),(18,49,316,244,198,341),(19,50,317,245,199,342),(20,51,318,246,200,343),(21,52,319,247,201,344),(22,53,320,248,202,345),(23,54,321,249,203,346),(24,55,322,250,204,347),(25,56,323,251,171,348),(26,57,324,252,172,349),(27,58,325,253,173,350),(28,59,326,254,174,351),(29,60,327,255,175,352),(30,61,328,256,176,353),(31,62,329,257,177,354),(32,63,330,258,178,355),(33,64,331,259,179,356),(34,65,332,260,180,357),(69,217,123,382,301,160),(70,218,124,383,302,161),(71,219,125,384,303,162),(72,220,126,385,304,163),(73,221,127,386,305,164),(74,222,128,387,306,165),(75,223,129,388,273,166),(76,224,130,389,274,167),(77,225,131,390,275,168),(78,226,132,391,276,169),(79,227,133,392,277,170),(80,228,134,393,278,137),(81,229,135,394,279,138),(82,230,136,395,280,139),(83,231,103,396,281,140),(84,232,104,397,282,141),(85,233,105,398,283,142),(86,234,106,399,284,143),(87,235,107,400,285,144),(88,236,108,401,286,145),(89,237,109,402,287,146),(90,238,110,403,288,147),(91,205,111,404,289,148),(92,206,112,405,290,149),(93,207,113,406,291,150),(94,208,114,407,292,151),(95,209,115,408,293,152),(96,210,116,375,294,153),(97,211,117,376,295,154),(98,212,118,377,296,155),(99,213,119,378,297,156),(100,214,120,379,298,157),(101,215,121,380,299,158),(102,216,122,381,300,159)], [(1,221,261,305),(2,222,262,306),(3,223,263,273),(4,224,264,274),(5,225,265,275),(6,226,266,276),(7,227,267,277),(8,228,268,278),(9,229,269,279),(10,230,270,280),(11,231,271,281),(12,232,272,282),(13,233,239,283),(14,234,240,284),(15,235,241,285),(16,236,242,286),(17,237,243,287),(18,238,244,288),(19,205,245,289),(20,206,246,290),(21,207,247,291),(22,208,248,292),(23,209,249,293),(24,210,250,294),(25,211,251,295),(26,212,252,296),(27,213,253,297),(28,214,254,298),(29,215,255,299),(30,216,256,300),(31,217,257,301),(32,218,258,302),(33,219,259,303),(34,220,260,304),(35,76,184,389),(36,77,185,390),(37,78,186,391),(38,79,187,392),(39,80,188,393),(40,81,189,394),(41,82,190,395),(42,83,191,396),(43,84,192,397),(44,85,193,398),(45,86,194,399),(46,87,195,400),(47,88,196,401),(48,89,197,402),(49,90,198,403),(50,91,199,404),(51,92,200,405),(52,93,201,406),(53,94,202,407),(54,95,203,408),(55,96,204,375),(56,97,171,376),(57,98,172,377),(58,99,173,378),(59,100,174,379),(60,101,175,380),(61,102,176,381),(62,69,177,382),(63,70,178,383),(64,71,179,384),(65,72,180,385),(66,73,181,386),(67,74,182,387),(68,75,183,388),(103,309,140,368),(104,310,141,369),(105,311,142,370),(106,312,143,371),(107,313,144,372),(108,314,145,373),(109,315,146,374),(110,316,147,341),(111,317,148,342),(112,318,149,343),(113,319,150,344),(114,320,151,345),(115,321,152,346),(116,322,153,347),(117,323,154,348),(118,324,155,349),(119,325,156,350),(120,326,157,351),(121,327,158,352),(122,328,159,353),(123,329,160,354),(124,330,161,355),(125,331,162,356),(126,332,163,357),(127,333,164,358),(128,334,165,359),(129,335,166,360),(130,336,167,361),(131,337,168,362),(132,338,169,363),(133,339,170,364),(134,340,137,365),(135,307,138,366),(136,308,139,367)]])
204 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 17A | ··· | 17P | 34A | ··· | 34AV | 51A | ··· | 51P | 68A | ··· | 68BL | 102A | ··· | 102AV |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 17 | ··· | 17 | 34 | ··· | 34 | 51 | ··· | 51 | 68 | ··· | 68 | 102 | ··· | 102 |
| size | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
204 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C4 | C17 | C34 | C34 | C68 | S3 | Dic3 | D6 | S3×C17 | Dic3×C17 | S3×C34 |
| kernel | Dic3×C34 | Dic3×C17 | C2×C102 | C102 | C2×Dic3 | Dic3 | C2×C6 | C6 | C2×C34 | C34 | C34 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 16 | 32 | 16 | 64 | 1 | 2 | 1 | 16 | 32 | 16 |
Matrix representation of Dic3×C34 ►in GL3(𝔽409) generated by
| 408 | 0 | 0 |
| 0 | 262 | 0 |
| 0 | 0 | 262 |
| 1 | 0 | 0 |
| 0 | 1 | 408 |
| 0 | 1 | 0 |
| 408 | 0 | 0 |
| 0 | 189 | 120 |
| 0 | 309 | 220 |
G:=sub<GL(3,GF(409))| [408,0,0,0,262,0,0,0,262],[1,0,0,0,1,1,0,408,0],[408,0,0,0,189,309,0,120,220] >;
Dic3×C34 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_{34} % in TeX
G:=Group("Dic3xC34"); // GroupNames label
G:=SmallGroup(408,23);
// by ID
G=gap.SmallGroup(408,23);
# by ID
G:=PCGroup([5,-2,-2,-17,-2,-3,340,6804]);
// Polycyclic
G:=Group<a,b,c|a^34=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export