direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic3×C35, C3⋊C140, C6.C70, C21⋊3C20, C15⋊5C28, C105⋊11C4, C70.4S3, C210.7C2, C30.3C14, C42.3C10, C2.(S3×C35), C14.2(C5×S3), C10.2(S3×C7), SmallGroup(420,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — Dic3×C35 |
Generators and relations for Dic3×C35
G = < a,b,c | a35=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of Dic3×C35
►Regular action on 420 points
Generators in S420
(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)(386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420)
(1 123 276 355 292 240)(2 124 277 356 293 241)(3 125 278 357 294 242)(4 126 279 358 295 243)(5 127 280 359 296 244)(6 128 246 360 297 245)(7 129 247 361 298 211)(8 130 248 362 299 212)(9 131 249 363 300 213)(10 132 250 364 301 214)(11 133 251 365 302 215)(12 134 252 366 303 216)(13 135 253 367 304 217)(14 136 254 368 305 218)(15 137 255 369 306 219)(16 138 256 370 307 220)(17 139 257 371 308 221)(18 140 258 372 309 222)(19 106 259 373 310 223)(20 107 260 374 311 224)(21 108 261 375 312 225)(22 109 262 376 313 226)(23 110 263 377 314 227)(24 111 264 378 315 228)(25 112 265 379 281 229)(26 113 266 380 282 230)(27 114 267 381 283 231)(28 115 268 382 284 232)(29 116 269 383 285 233)(30 117 270 384 286 234)(31 118 271 385 287 235)(32 119 272 351 288 236)(33 120 273 352 289 237)(34 121 274 353 290 238)(35 122 275 354 291 239)(36 199 396 341 100 171)(37 200 397 342 101 172)(38 201 398 343 102 173)(39 202 399 344 103 174)(40 203 400 345 104 175)(41 204 401 346 105 141)(42 205 402 347 71 142)(43 206 403 348 72 143)(44 207 404 349 73 144)(45 208 405 350 74 145)(46 209 406 316 75 146)(47 210 407 317 76 147)(48 176 408 318 77 148)(49 177 409 319 78 149)(50 178 410 320 79 150)(51 179 411 321 80 151)(52 180 412 322 81 152)(53 181 413 323 82 153)(54 182 414 324 83 154)(55 183 415 325 84 155)(56 184 416 326 85 156)(57 185 417 327 86 157)(58 186 418 328 87 158)(59 187 419 329 88 159)(60 188 420 330 89 160)(61 189 386 331 90 161)(62 190 387 332 91 162)(63 191 388 333 92 163)(64 192 389 334 93 164)(65 193 390 335 94 165)(66 194 391 336 95 166)(67 195 392 337 96 167)(68 196 393 338 97 168)(69 197 394 339 98 169)(70 198 395 340 99 170)
(1 196 355 97)(2 197 356 98)(3 198 357 99)(4 199 358 100)(5 200 359 101)(6 201 360 102)(7 202 361 103)(8 203 362 104)(9 204 363 105)(10 205 364 71)(11 206 365 72)(12 207 366 73)(13 208 367 74)(14 209 368 75)(15 210 369 76)(16 176 370 77)(17 177 371 78)(18 178 372 79)(19 179 373 80)(20 180 374 81)(21 181 375 82)(22 182 376 83)(23 183 377 84)(24 184 378 85)(25 185 379 86)(26 186 380 87)(27 187 381 88)(28 188 382 89)(29 189 383 90)(30 190 384 91)(31 191 385 92)(32 192 351 93)(33 193 352 94)(34 194 353 95)(35 195 354 96)(36 295 341 126)(37 296 342 127)(38 297 343 128)(39 298 344 129)(40 299 345 130)(41 300 346 131)(42 301 347 132)(43 302 348 133)(44 303 349 134)(45 304 350 135)(46 305 316 136)(47 306 317 137)(48 307 318 138)(49 308 319 139)(50 309 320 140)(51 310 321 106)(52 311 322 107)(53 312 323 108)(54 313 324 109)(55 314 325 110)(56 315 326 111)(57 281 327 112)(58 282 328 113)(59 283 329 114)(60 284 330 115)(61 285 331 116)(62 286 332 117)(63 287 333 118)(64 288 334 119)(65 289 335 120)(66 290 336 121)(67 291 337 122)(68 292 338 123)(69 293 339 124)(70 294 340 125)(141 213 401 249)(142 214 402 250)(143 215 403 251)(144 216 404 252)(145 217 405 253)(146 218 406 254)(147 219 407 255)(148 220 408 256)(149 221 409 257)(150 222 410 258)(151 223 411 259)(152 224 412 260)(153 225 413 261)(154 226 414 262)(155 227 415 263)(156 228 416 264)(157 229 417 265)(158 230 418 266)(159 231 419 267)(160 232 420 268)(161 233 386 269)(162 234 387 270)(163 235 388 271)(164 236 389 272)(165 237 390 273)(166 238 391 274)(167 239 392 275)(168 240 393 276)(169 241 394 277)(170 242 395 278)(171 243 396 279)(172 244 397 280)(173 245 398 246)(174 211 399 247)(175 212 400 248)
G:=sub<Sym(420)| (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)(386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420), (1,123,276,355,292,240)(2,124,277,356,293,241)(3,125,278,357,294,242)(4,126,279,358,295,243)(5,127,280,359,296,244)(6,128,246,360,297,245)(7,129,247,361,298,211)(8,130,248,362,299,212)(9,131,249,363,300,213)(10,132,250,364,301,214)(11,133,251,365,302,215)(12,134,252,366,303,216)(13,135,253,367,304,217)(14,136,254,368,305,218)(15,137,255,369,306,219)(16,138,256,370,307,220)(17,139,257,371,308,221)(18,140,258,372,309,222)(19,106,259,373,310,223)(20,107,260,374,311,224)(21,108,261,375,312,225)(22,109,262,376,313,226)(23,110,263,377,314,227)(24,111,264,378,315,228)(25,112,265,379,281,229)(26,113,266,380,282,230)(27,114,267,381,283,231)(28,115,268,382,284,232)(29,116,269,383,285,233)(30,117,270,384,286,234)(31,118,271,385,287,235)(32,119,272,351,288,236)(33,120,273,352,289,237)(34,121,274,353,290,238)(35,122,275,354,291,239)(36,199,396,341,100,171)(37,200,397,342,101,172)(38,201,398,343,102,173)(39,202,399,344,103,174)(40,203,400,345,104,175)(41,204,401,346,105,141)(42,205,402,347,71,142)(43,206,403,348,72,143)(44,207,404,349,73,144)(45,208,405,350,74,145)(46,209,406,316,75,146)(47,210,407,317,76,147)(48,176,408,318,77,148)(49,177,409,319,78,149)(50,178,410,320,79,150)(51,179,411,321,80,151)(52,180,412,322,81,152)(53,181,413,323,82,153)(54,182,414,324,83,154)(55,183,415,325,84,155)(56,184,416,326,85,156)(57,185,417,327,86,157)(58,186,418,328,87,158)(59,187,419,329,88,159)(60,188,420,330,89,160)(61,189,386,331,90,161)(62,190,387,332,91,162)(63,191,388,333,92,163)(64,192,389,334,93,164)(65,193,390,335,94,165)(66,194,391,336,95,166)(67,195,392,337,96,167)(68,196,393,338,97,168)(69,197,394,339,98,169)(70,198,395,340,99,170), (1,196,355,97)(2,197,356,98)(3,198,357,99)(4,199,358,100)(5,200,359,101)(6,201,360,102)(7,202,361,103)(8,203,362,104)(9,204,363,105)(10,205,364,71)(11,206,365,72)(12,207,366,73)(13,208,367,74)(14,209,368,75)(15,210,369,76)(16,176,370,77)(17,177,371,78)(18,178,372,79)(19,179,373,80)(20,180,374,81)(21,181,375,82)(22,182,376,83)(23,183,377,84)(24,184,378,85)(25,185,379,86)(26,186,380,87)(27,187,381,88)(28,188,382,89)(29,189,383,90)(30,190,384,91)(31,191,385,92)(32,192,351,93)(33,193,352,94)(34,194,353,95)(35,195,354,96)(36,295,341,126)(37,296,342,127)(38,297,343,128)(39,298,344,129)(40,299,345,130)(41,300,346,131)(42,301,347,132)(43,302,348,133)(44,303,349,134)(45,304,350,135)(46,305,316,136)(47,306,317,137)(48,307,318,138)(49,308,319,139)(50,309,320,140)(51,310,321,106)(52,311,322,107)(53,312,323,108)(54,313,324,109)(55,314,325,110)(56,315,326,111)(57,281,327,112)(58,282,328,113)(59,283,329,114)(60,284,330,115)(61,285,331,116)(62,286,332,117)(63,287,333,118)(64,288,334,119)(65,289,335,120)(66,290,336,121)(67,291,337,122)(68,292,338,123)(69,293,339,124)(70,294,340,125)(141,213,401,249)(142,214,402,250)(143,215,403,251)(144,216,404,252)(145,217,405,253)(146,218,406,254)(147,219,407,255)(148,220,408,256)(149,221,409,257)(150,222,410,258)(151,223,411,259)(152,224,412,260)(153,225,413,261)(154,226,414,262)(155,227,415,263)(156,228,416,264)(157,229,417,265)(158,230,418,266)(159,231,419,267)(160,232,420,268)(161,233,386,269)(162,234,387,270)(163,235,388,271)(164,236,389,272)(165,237,390,273)(166,238,391,274)(167,239,392,275)(168,240,393,276)(169,241,394,277)(170,242,395,278)(171,243,396,279)(172,244,397,280)(173,245,398,246)(174,211,399,247)(175,212,400,248)>;
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)(386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420), (1,123,276,355,292,240)(2,124,277,356,293,241)(3,125,278,357,294,242)(4,126,279,358,295,243)(5,127,280,359,296,244)(6,128,246,360,297,245)(7,129,247,361,298,211)(8,130,248,362,299,212)(9,131,249,363,300,213)(10,132,250,364,301,214)(11,133,251,365,302,215)(12,134,252,366,303,216)(13,135,253,367,304,217)(14,136,254,368,305,218)(15,137,255,369,306,219)(16,138,256,370,307,220)(17,139,257,371,308,221)(18,140,258,372,309,222)(19,106,259,373,310,223)(20,107,260,374,311,224)(21,108,261,375,312,225)(22,109,262,376,313,226)(23,110,263,377,314,227)(24,111,264,378,315,228)(25,112,265,379,281,229)(26,113,266,380,282,230)(27,114,267,381,283,231)(28,115,268,382,284,232)(29,116,269,383,285,233)(30,117,270,384,286,234)(31,118,271,385,287,235)(32,119,272,351,288,236)(33,120,273,352,289,237)(34,121,274,353,290,238)(35,122,275,354,291,239)(36,199,396,341,100,171)(37,200,397,342,101,172)(38,201,398,343,102,173)(39,202,399,344,103,174)(40,203,400,345,104,175)(41,204,401,346,105,141)(42,205,402,347,71,142)(43,206,403,348,72,143)(44,207,404,349,73,144)(45,208,405,350,74,145)(46,209,406,316,75,146)(47,210,407,317,76,147)(48,176,408,318,77,148)(49,177,409,319,78,149)(50,178,410,320,79,150)(51,179,411,321,80,151)(52,180,412,322,81,152)(53,181,413,323,82,153)(54,182,414,324,83,154)(55,183,415,325,84,155)(56,184,416,326,85,156)(57,185,417,327,86,157)(58,186,418,328,87,158)(59,187,419,329,88,159)(60,188,420,330,89,160)(61,189,386,331,90,161)(62,190,387,332,91,162)(63,191,388,333,92,163)(64,192,389,334,93,164)(65,193,390,335,94,165)(66,194,391,336,95,166)(67,195,392,337,96,167)(68,196,393,338,97,168)(69,197,394,339,98,169)(70,198,395,340,99,170), (1,196,355,97)(2,197,356,98)(3,198,357,99)(4,199,358,100)(5,200,359,101)(6,201,360,102)(7,202,361,103)(8,203,362,104)(9,204,363,105)(10,205,364,71)(11,206,365,72)(12,207,366,73)(13,208,367,74)(14,209,368,75)(15,210,369,76)(16,176,370,77)(17,177,371,78)(18,178,372,79)(19,179,373,80)(20,180,374,81)(21,181,375,82)(22,182,376,83)(23,183,377,84)(24,184,378,85)(25,185,379,86)(26,186,380,87)(27,187,381,88)(28,188,382,89)(29,189,383,90)(30,190,384,91)(31,191,385,92)(32,192,351,93)(33,193,352,94)(34,194,353,95)(35,195,354,96)(36,295,341,126)(37,296,342,127)(38,297,343,128)(39,298,344,129)(40,299,345,130)(41,300,346,131)(42,301,347,132)(43,302,348,133)(44,303,349,134)(45,304,350,135)(46,305,316,136)(47,306,317,137)(48,307,318,138)(49,308,319,139)(50,309,320,140)(51,310,321,106)(52,311,322,107)(53,312,323,108)(54,313,324,109)(55,314,325,110)(56,315,326,111)(57,281,327,112)(58,282,328,113)(59,283,329,114)(60,284,330,115)(61,285,331,116)(62,286,332,117)(63,287,333,118)(64,288,334,119)(65,289,335,120)(66,290,336,121)(67,291,337,122)(68,292,338,123)(69,293,339,124)(70,294,340,125)(141,213,401,249)(142,214,402,250)(143,215,403,251)(144,216,404,252)(145,217,405,253)(146,218,406,254)(147,219,407,255)(148,220,408,256)(149,221,409,257)(150,222,410,258)(151,223,411,259)(152,224,412,260)(153,225,413,261)(154,226,414,262)(155,227,415,263)(156,228,416,264)(157,229,417,265)(158,230,418,266)(159,231,419,267)(160,232,420,268)(161,233,386,269)(162,234,387,270)(163,235,388,271)(164,236,389,272)(165,237,390,273)(166,238,391,274)(167,239,392,275)(168,240,393,276)(169,241,394,277)(170,242,395,278)(171,243,396,279)(172,244,397,280)(173,245,398,246)(174,211,399,247)(175,212,400,248) );
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),(386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420)], [(1,123,276,355,292,240),(2,124,277,356,293,241),(3,125,278,357,294,242),(4,126,279,358,295,243),(5,127,280,359,296,244),(6,128,246,360,297,245),(7,129,247,361,298,211),(8,130,248,362,299,212),(9,131,249,363,300,213),(10,132,250,364,301,214),(11,133,251,365,302,215),(12,134,252,366,303,216),(13,135,253,367,304,217),(14,136,254,368,305,218),(15,137,255,369,306,219),(16,138,256,370,307,220),(17,139,257,371,308,221),(18,140,258,372,309,222),(19,106,259,373,310,223),(20,107,260,374,311,224),(21,108,261,375,312,225),(22,109,262,376,313,226),(23,110,263,377,314,227),(24,111,264,378,315,228),(25,112,265,379,281,229),(26,113,266,380,282,230),(27,114,267,381,283,231),(28,115,268,382,284,232),(29,116,269,383,285,233),(30,117,270,384,286,234),(31,118,271,385,287,235),(32,119,272,351,288,236),(33,120,273,352,289,237),(34,121,274,353,290,238),(35,122,275,354,291,239),(36,199,396,341,100,171),(37,200,397,342,101,172),(38,201,398,343,102,173),(39,202,399,344,103,174),(40,203,400,345,104,175),(41,204,401,346,105,141),(42,205,402,347,71,142),(43,206,403,348,72,143),(44,207,404,349,73,144),(45,208,405,350,74,145),(46,209,406,316,75,146),(47,210,407,317,76,147),(48,176,408,318,77,148),(49,177,409,319,78,149),(50,178,410,320,79,150),(51,179,411,321,80,151),(52,180,412,322,81,152),(53,181,413,323,82,153),(54,182,414,324,83,154),(55,183,415,325,84,155),(56,184,416,326,85,156),(57,185,417,327,86,157),(58,186,418,328,87,158),(59,187,419,329,88,159),(60,188,420,330,89,160),(61,189,386,331,90,161),(62,190,387,332,91,162),(63,191,388,333,92,163),(64,192,389,334,93,164),(65,193,390,335,94,165),(66,194,391,336,95,166),(67,195,392,337,96,167),(68,196,393,338,97,168),(69,197,394,339,98,169),(70,198,395,340,99,170)], [(1,196,355,97),(2,197,356,98),(3,198,357,99),(4,199,358,100),(5,200,359,101),(6,201,360,102),(7,202,361,103),(8,203,362,104),(9,204,363,105),(10,205,364,71),(11,206,365,72),(12,207,366,73),(13,208,367,74),(14,209,368,75),(15,210,369,76),(16,176,370,77),(17,177,371,78),(18,178,372,79),(19,179,373,80),(20,180,374,81),(21,181,375,82),(22,182,376,83),(23,183,377,84),(24,184,378,85),(25,185,379,86),(26,186,380,87),(27,187,381,88),(28,188,382,89),(29,189,383,90),(30,190,384,91),(31,191,385,92),(32,192,351,93),(33,193,352,94),(34,194,353,95),(35,195,354,96),(36,295,341,126),(37,296,342,127),(38,297,343,128),(39,298,344,129),(40,299,345,130),(41,300,346,131),(42,301,347,132),(43,302,348,133),(44,303,349,134),(45,304,350,135),(46,305,316,136),(47,306,317,137),(48,307,318,138),(49,308,319,139),(50,309,320,140),(51,310,321,106),(52,311,322,107),(53,312,323,108),(54,313,324,109),(55,314,325,110),(56,315,326,111),(57,281,327,112),(58,282,328,113),(59,283,329,114),(60,284,330,115),(61,285,331,116),(62,286,332,117),(63,287,333,118),(64,288,334,119),(65,289,335,120),(66,290,336,121),(67,291,337,122),(68,292,338,123),(69,293,339,124),(70,294,340,125),(141,213,401,249),(142,214,402,250),(143,215,403,251),(144,216,404,252),(145,217,405,253),(146,218,406,254),(147,219,407,255),(148,220,408,256),(149,221,409,257),(150,222,410,258),(151,223,411,259),(152,224,412,260),(153,225,413,261),(154,226,414,262),(155,227,415,263),(156,228,416,264),(157,229,417,265),(158,230,418,266),(159,231,419,267),(160,232,420,268),(161,233,386,269),(162,234,387,270),(163,235,388,271),(164,236,389,272),(165,237,390,273),(166,238,391,274),(167,239,392,275),(168,240,393,276),(169,241,394,277),(170,242,395,278),(171,243,396,279),(172,244,397,280),(173,245,398,246),(174,211,399,247),(175,212,400,248)]])
210 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6 | 7A | ··· | 7F | 10A | 10B | 10C | 10D | 14A | ··· | 14F | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 21A | ··· | 21F | 28A | ··· | 28L | 30A | 30B | 30C | 30D | 35A | ··· | 35X | 42A | ··· | 42F | 70A | ··· | 70X | 105A | ··· | 105X | 140A | ··· | 140AV | 210A | ··· | 210X |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 7 | ··· | 7 | 10 | 10 | 10 | 10 | 14 | ··· | 14 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 21 | ··· | 21 | 28 | ··· | 28 | 30 | 30 | 30 | 30 | 35 | ··· | 35 | 42 | ··· | 42 | 70 | ··· | 70 | 105 | ··· | 105 | 140 | ··· | 140 | 210 | ··· | 210 |
| size | 1 | 1 | 2 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 2 | ··· | 2 |
210 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||||||||
| image | C1 | C2 | C4 | C5 | C7 | C10 | C14 | C20 | C28 | C35 | C70 | C140 | S3 | Dic3 | C5×S3 | S3×C7 | C5×Dic3 | C7×Dic3 | S3×C35 | Dic3×C35 |
| kernel | Dic3×C35 | C210 | C105 | C7×Dic3 | C5×Dic3 | C42 | C30 | C21 | C15 | Dic3 | C6 | C3 | C70 | C35 | C14 | C10 | C7 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 6 | 4 | 6 | 8 | 12 | 24 | 24 | 48 | 1 | 1 | 4 | 6 | 4 | 6 | 24 | 24 |
Matrix representation of Dic3×C35 ►in GL2(𝔽421) generated by
| 291 | 0 |
| 0 | 291 |
| 0 | 420 |
| 1 | 1 |
| 43 | 66 |
| 23 | 378 |
G:=sub<GL(2,GF(421))| [291,0,0,291],[0,1,420,1],[43,23,66,378] >;
Dic3×C35 in GAP, Magma, Sage, TeX
{\rm Dic}_3\times C_{35} % in TeX
G:=Group("Dic3xC35"); // GroupNames label
G:=SmallGroup(420,8);
// by ID
G=gap.SmallGroup(420,8);
# by ID
G:=PCGroup([5,-2,-5,-7,-2,-3,350,7004]);
// Polycyclic
G:=Group<a,b,c|a^35=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export