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## G = C3×Dic35order 420 = 22·3·5·7

### Direct product of C3 and Dic35

Aliases: C3×Dic35, C359C12, C1056C4, C70.5C6, C6.2D35, C30.2D7, C42.2D5, C154Dic7, C212Dic5, C210.2C2, C10.(C3×D7), C2.(C3×D35), C52(C3×Dic7), C73(C3×Dic5), C14.3(C3×D5), SmallGroup(420,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C35 — C3×Dic35
 Chief series C1 — C7 — C35 — C70 — C210 — C3×Dic35
 Lower central C35 — C3×Dic35
 Upper central C1 — C6

Generators and relations for C3×Dic35
G = < a,b,c | a3=b70=1, c2=b35, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C3×Dic35
Regular action on 420 points
Generators in S420
(1 158 83)(2 159 84)(3 160 85)(4 161 86)(5 162 87)(6 163 88)(7 164 89)(8 165 90)(9 166 91)(10 167 92)(11 168 93)(12 169 94)(13 170 95)(14 171 96)(15 172 97)(16 173 98)(17 174 99)(18 175 100)(19 176 101)(20 177 102)(21 178 103)(22 179 104)(23 180 105)(24 181 106)(25 182 107)(26 183 108)(27 184 109)(28 185 110)(29 186 111)(30 187 112)(31 188 113)(32 189 114)(33 190 115)(34 191 116)(35 192 117)(36 193 118)(37 194 119)(38 195 120)(39 196 121)(40 197 122)(41 198 123)(42 199 124)(43 200 125)(44 201 126)(45 202 127)(46 203 128)(47 204 129)(48 205 130)(49 206 131)(50 207 132)(51 208 133)(52 209 134)(53 210 135)(54 141 136)(55 142 137)(56 143 138)(57 144 139)(58 145 140)(59 146 71)(60 147 72)(61 148 73)(62 149 74)(63 150 75)(64 151 76)(65 152 77)(66 153 78)(67 154 79)(68 155 80)(69 156 81)(70 157 82)(211 414 316)(212 415 317)(213 416 318)(214 417 319)(215 418 320)(216 419 321)(217 420 322)(218 351 323)(219 352 324)(220 353 325)(221 354 326)(222 355 327)(223 356 328)(224 357 329)(225 358 330)(226 359 331)(227 360 332)(228 361 333)(229 362 334)(230 363 335)(231 364 336)(232 365 337)(233 366 338)(234 367 339)(235 368 340)(236 369 341)(237 370 342)(238 371 343)(239 372 344)(240 373 345)(241 374 346)(242 375 347)(243 376 348)(244 377 349)(245 378 350)(246 379 281)(247 380 282)(248 381 283)(249 382 284)(250 383 285)(251 384 286)(252 385 287)(253 386 288)(254 387 289)(255 388 290)(256 389 291)(257 390 292)(258 391 293)(259 392 294)(260 393 295)(261 394 296)(262 395 297)(263 396 298)(264 397 299)(265 398 300)(266 399 301)(267 400 302)(268 401 303)(269 402 304)(270 403 305)(271 404 306)(272 405 307)(273 406 308)(274 407 309)(275 408 310)(276 409 311)(277 410 312)(278 411 313)(279 412 314)(280 413 315)
(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)
(1 225 36 260)(2 224 37 259)(3 223 38 258)(4 222 39 257)(5 221 40 256)(6 220 41 255)(7 219 42 254)(8 218 43 253)(9 217 44 252)(10 216 45 251)(11 215 46 250)(12 214 47 249)(13 213 48 248)(14 212 49 247)(15 211 50 246)(16 280 51 245)(17 279 52 244)(18 278 53 243)(19 277 54 242)(20 276 55 241)(21 275 56 240)(22 274 57 239)(23 273 58 238)(24 272 59 237)(25 271 60 236)(26 270 61 235)(27 269 62 234)(28 268 63 233)(29 267 64 232)(30 266 65 231)(31 265 66 230)(32 264 67 229)(33 263 68 228)(34 262 69 227)(35 261 70 226)(71 342 106 307)(72 341 107 306)(73 340 108 305)(74 339 109 304)(75 338 110 303)(76 337 111 302)(77 336 112 301)(78 335 113 300)(79 334 114 299)(80 333 115 298)(81 332 116 297)(82 331 117 296)(83 330 118 295)(84 329 119 294)(85 328 120 293)(86 327 121 292)(87 326 122 291)(88 325 123 290)(89 324 124 289)(90 323 125 288)(91 322 126 287)(92 321 127 286)(93 320 128 285)(94 319 129 284)(95 318 130 283)(96 317 131 282)(97 316 132 281)(98 315 133 350)(99 314 134 349)(100 313 135 348)(101 312 136 347)(102 311 137 346)(103 310 138 345)(104 309 139 344)(105 308 140 343)(141 375 176 410)(142 374 177 409)(143 373 178 408)(144 372 179 407)(145 371 180 406)(146 370 181 405)(147 369 182 404)(148 368 183 403)(149 367 184 402)(150 366 185 401)(151 365 186 400)(152 364 187 399)(153 363 188 398)(154 362 189 397)(155 361 190 396)(156 360 191 395)(157 359 192 394)(158 358 193 393)(159 357 194 392)(160 356 195 391)(161 355 196 390)(162 354 197 389)(163 353 198 388)(164 352 199 387)(165 351 200 386)(166 420 201 385)(167 419 202 384)(168 418 203 383)(169 417 204 382)(170 416 205 381)(171 415 206 380)(172 414 207 379)(173 413 208 378)(174 412 209 377)(175 411 210 376)

G:=sub<Sym(420)| (1,158,83)(2,159,84)(3,160,85)(4,161,86)(5,162,87)(6,163,88)(7,164,89)(8,165,90)(9,166,91)(10,167,92)(11,168,93)(12,169,94)(13,170,95)(14,171,96)(15,172,97)(16,173,98)(17,174,99)(18,175,100)(19,176,101)(20,177,102)(21,178,103)(22,179,104)(23,180,105)(24,181,106)(25,182,107)(26,183,108)(27,184,109)(28,185,110)(29,186,111)(30,187,112)(31,188,113)(32,189,114)(33,190,115)(34,191,116)(35,192,117)(36,193,118)(37,194,119)(38,195,120)(39,196,121)(40,197,122)(41,198,123)(42,199,124)(43,200,125)(44,201,126)(45,202,127)(46,203,128)(47,204,129)(48,205,130)(49,206,131)(50,207,132)(51,208,133)(52,209,134)(53,210,135)(54,141,136)(55,142,137)(56,143,138)(57,144,139)(58,145,140)(59,146,71)(60,147,72)(61,148,73)(62,149,74)(63,150,75)(64,151,76)(65,152,77)(66,153,78)(67,154,79)(68,155,80)(69,156,81)(70,157,82)(211,414,316)(212,415,317)(213,416,318)(214,417,319)(215,418,320)(216,419,321)(217,420,322)(218,351,323)(219,352,324)(220,353,325)(221,354,326)(222,355,327)(223,356,328)(224,357,329)(225,358,330)(226,359,331)(227,360,332)(228,361,333)(229,362,334)(230,363,335)(231,364,336)(232,365,337)(233,366,338)(234,367,339)(235,368,340)(236,369,341)(237,370,342)(238,371,343)(239,372,344)(240,373,345)(241,374,346)(242,375,347)(243,376,348)(244,377,349)(245,378,350)(246,379,281)(247,380,282)(248,381,283)(249,382,284)(250,383,285)(251,384,286)(252,385,287)(253,386,288)(254,387,289)(255,388,290)(256,389,291)(257,390,292)(258,391,293)(259,392,294)(260,393,295)(261,394,296)(262,395,297)(263,396,298)(264,397,299)(265,398,300)(266,399,301)(267,400,302)(268,401,303)(269,402,304)(270,403,305)(271,404,306)(272,405,307)(273,406,308)(274,407,309)(275,408,310)(276,409,311)(277,410,312)(278,411,313)(279,412,314)(280,413,315), (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), (1,225,36,260)(2,224,37,259)(3,223,38,258)(4,222,39,257)(5,221,40,256)(6,220,41,255)(7,219,42,254)(8,218,43,253)(9,217,44,252)(10,216,45,251)(11,215,46,250)(12,214,47,249)(13,213,48,248)(14,212,49,247)(15,211,50,246)(16,280,51,245)(17,279,52,244)(18,278,53,243)(19,277,54,242)(20,276,55,241)(21,275,56,240)(22,274,57,239)(23,273,58,238)(24,272,59,237)(25,271,60,236)(26,270,61,235)(27,269,62,234)(28,268,63,233)(29,267,64,232)(30,266,65,231)(31,265,66,230)(32,264,67,229)(33,263,68,228)(34,262,69,227)(35,261,70,226)(71,342,106,307)(72,341,107,306)(73,340,108,305)(74,339,109,304)(75,338,110,303)(76,337,111,302)(77,336,112,301)(78,335,113,300)(79,334,114,299)(80,333,115,298)(81,332,116,297)(82,331,117,296)(83,330,118,295)(84,329,119,294)(85,328,120,293)(86,327,121,292)(87,326,122,291)(88,325,123,290)(89,324,124,289)(90,323,125,288)(91,322,126,287)(92,321,127,286)(93,320,128,285)(94,319,129,284)(95,318,130,283)(96,317,131,282)(97,316,132,281)(98,315,133,350)(99,314,134,349)(100,313,135,348)(101,312,136,347)(102,311,137,346)(103,310,138,345)(104,309,139,344)(105,308,140,343)(141,375,176,410)(142,374,177,409)(143,373,178,408)(144,372,179,407)(145,371,180,406)(146,370,181,405)(147,369,182,404)(148,368,183,403)(149,367,184,402)(150,366,185,401)(151,365,186,400)(152,364,187,399)(153,363,188,398)(154,362,189,397)(155,361,190,396)(156,360,191,395)(157,359,192,394)(158,358,193,393)(159,357,194,392)(160,356,195,391)(161,355,196,390)(162,354,197,389)(163,353,198,388)(164,352,199,387)(165,351,200,386)(166,420,201,385)(167,419,202,384)(168,418,203,383)(169,417,204,382)(170,416,205,381)(171,415,206,380)(172,414,207,379)(173,413,208,378)(174,412,209,377)(175,411,210,376)>;

G:=Group( (1,158,83)(2,159,84)(3,160,85)(4,161,86)(5,162,87)(6,163,88)(7,164,89)(8,165,90)(9,166,91)(10,167,92)(11,168,93)(12,169,94)(13,170,95)(14,171,96)(15,172,97)(16,173,98)(17,174,99)(18,175,100)(19,176,101)(20,177,102)(21,178,103)(22,179,104)(23,180,105)(24,181,106)(25,182,107)(26,183,108)(27,184,109)(28,185,110)(29,186,111)(30,187,112)(31,188,113)(32,189,114)(33,190,115)(34,191,116)(35,192,117)(36,193,118)(37,194,119)(38,195,120)(39,196,121)(40,197,122)(41,198,123)(42,199,124)(43,200,125)(44,201,126)(45,202,127)(46,203,128)(47,204,129)(48,205,130)(49,206,131)(50,207,132)(51,208,133)(52,209,134)(53,210,135)(54,141,136)(55,142,137)(56,143,138)(57,144,139)(58,145,140)(59,146,71)(60,147,72)(61,148,73)(62,149,74)(63,150,75)(64,151,76)(65,152,77)(66,153,78)(67,154,79)(68,155,80)(69,156,81)(70,157,82)(211,414,316)(212,415,317)(213,416,318)(214,417,319)(215,418,320)(216,419,321)(217,420,322)(218,351,323)(219,352,324)(220,353,325)(221,354,326)(222,355,327)(223,356,328)(224,357,329)(225,358,330)(226,359,331)(227,360,332)(228,361,333)(229,362,334)(230,363,335)(231,364,336)(232,365,337)(233,366,338)(234,367,339)(235,368,340)(236,369,341)(237,370,342)(238,371,343)(239,372,344)(240,373,345)(241,374,346)(242,375,347)(243,376,348)(244,377,349)(245,378,350)(246,379,281)(247,380,282)(248,381,283)(249,382,284)(250,383,285)(251,384,286)(252,385,287)(253,386,288)(254,387,289)(255,388,290)(256,389,291)(257,390,292)(258,391,293)(259,392,294)(260,393,295)(261,394,296)(262,395,297)(263,396,298)(264,397,299)(265,398,300)(266,399,301)(267,400,302)(268,401,303)(269,402,304)(270,403,305)(271,404,306)(272,405,307)(273,406,308)(274,407,309)(275,408,310)(276,409,311)(277,410,312)(278,411,313)(279,412,314)(280,413,315), (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), (1,225,36,260)(2,224,37,259)(3,223,38,258)(4,222,39,257)(5,221,40,256)(6,220,41,255)(7,219,42,254)(8,218,43,253)(9,217,44,252)(10,216,45,251)(11,215,46,250)(12,214,47,249)(13,213,48,248)(14,212,49,247)(15,211,50,246)(16,280,51,245)(17,279,52,244)(18,278,53,243)(19,277,54,242)(20,276,55,241)(21,275,56,240)(22,274,57,239)(23,273,58,238)(24,272,59,237)(25,271,60,236)(26,270,61,235)(27,269,62,234)(28,268,63,233)(29,267,64,232)(30,266,65,231)(31,265,66,230)(32,264,67,229)(33,263,68,228)(34,262,69,227)(35,261,70,226)(71,342,106,307)(72,341,107,306)(73,340,108,305)(74,339,109,304)(75,338,110,303)(76,337,111,302)(77,336,112,301)(78,335,113,300)(79,334,114,299)(80,333,115,298)(81,332,116,297)(82,331,117,296)(83,330,118,295)(84,329,119,294)(85,328,120,293)(86,327,121,292)(87,326,122,291)(88,325,123,290)(89,324,124,289)(90,323,125,288)(91,322,126,287)(92,321,127,286)(93,320,128,285)(94,319,129,284)(95,318,130,283)(96,317,131,282)(97,316,132,281)(98,315,133,350)(99,314,134,349)(100,313,135,348)(101,312,136,347)(102,311,137,346)(103,310,138,345)(104,309,139,344)(105,308,140,343)(141,375,176,410)(142,374,177,409)(143,373,178,408)(144,372,179,407)(145,371,180,406)(146,370,181,405)(147,369,182,404)(148,368,183,403)(149,367,184,402)(150,366,185,401)(151,365,186,400)(152,364,187,399)(153,363,188,398)(154,362,189,397)(155,361,190,396)(156,360,191,395)(157,359,192,394)(158,358,193,393)(159,357,194,392)(160,356,195,391)(161,355,196,390)(162,354,197,389)(163,353,198,388)(164,352,199,387)(165,351,200,386)(166,420,201,385)(167,419,202,384)(168,418,203,383)(169,417,204,382)(170,416,205,381)(171,415,206,380)(172,414,207,379)(173,413,208,378)(174,412,209,377)(175,411,210,376) );

G=PermutationGroup([[(1,158,83),(2,159,84),(3,160,85),(4,161,86),(5,162,87),(6,163,88),(7,164,89),(8,165,90),(9,166,91),(10,167,92),(11,168,93),(12,169,94),(13,170,95),(14,171,96),(15,172,97),(16,173,98),(17,174,99),(18,175,100),(19,176,101),(20,177,102),(21,178,103),(22,179,104),(23,180,105),(24,181,106),(25,182,107),(26,183,108),(27,184,109),(28,185,110),(29,186,111),(30,187,112),(31,188,113),(32,189,114),(33,190,115),(34,191,116),(35,192,117),(36,193,118),(37,194,119),(38,195,120),(39,196,121),(40,197,122),(41,198,123),(42,199,124),(43,200,125),(44,201,126),(45,202,127),(46,203,128),(47,204,129),(48,205,130),(49,206,131),(50,207,132),(51,208,133),(52,209,134),(53,210,135),(54,141,136),(55,142,137),(56,143,138),(57,144,139),(58,145,140),(59,146,71),(60,147,72),(61,148,73),(62,149,74),(63,150,75),(64,151,76),(65,152,77),(66,153,78),(67,154,79),(68,155,80),(69,156,81),(70,157,82),(211,414,316),(212,415,317),(213,416,318),(214,417,319),(215,418,320),(216,419,321),(217,420,322),(218,351,323),(219,352,324),(220,353,325),(221,354,326),(222,355,327),(223,356,328),(224,357,329),(225,358,330),(226,359,331),(227,360,332),(228,361,333),(229,362,334),(230,363,335),(231,364,336),(232,365,337),(233,366,338),(234,367,339),(235,368,340),(236,369,341),(237,370,342),(238,371,343),(239,372,344),(240,373,345),(241,374,346),(242,375,347),(243,376,348),(244,377,349),(245,378,350),(246,379,281),(247,380,282),(248,381,283),(249,382,284),(250,383,285),(251,384,286),(252,385,287),(253,386,288),(254,387,289),(255,388,290),(256,389,291),(257,390,292),(258,391,293),(259,392,294),(260,393,295),(261,394,296),(262,395,297),(263,396,298),(264,397,299),(265,398,300),(266,399,301),(267,400,302),(268,401,303),(269,402,304),(270,403,305),(271,404,306),(272,405,307),(273,406,308),(274,407,309),(275,408,310),(276,409,311),(277,410,312),(278,411,313),(279,412,314),(280,413,315)], [(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)], [(1,225,36,260),(2,224,37,259),(3,223,38,258),(4,222,39,257),(5,221,40,256),(6,220,41,255),(7,219,42,254),(8,218,43,253),(9,217,44,252),(10,216,45,251),(11,215,46,250),(12,214,47,249),(13,213,48,248),(14,212,49,247),(15,211,50,246),(16,280,51,245),(17,279,52,244),(18,278,53,243),(19,277,54,242),(20,276,55,241),(21,275,56,240),(22,274,57,239),(23,273,58,238),(24,272,59,237),(25,271,60,236),(26,270,61,235),(27,269,62,234),(28,268,63,233),(29,267,64,232),(30,266,65,231),(31,265,66,230),(32,264,67,229),(33,263,68,228),(34,262,69,227),(35,261,70,226),(71,342,106,307),(72,341,107,306),(73,340,108,305),(74,339,109,304),(75,338,110,303),(76,337,111,302),(77,336,112,301),(78,335,113,300),(79,334,114,299),(80,333,115,298),(81,332,116,297),(82,331,117,296),(83,330,118,295),(84,329,119,294),(85,328,120,293),(86,327,121,292),(87,326,122,291),(88,325,123,290),(89,324,124,289),(90,323,125,288),(91,322,126,287),(92,321,127,286),(93,320,128,285),(94,319,129,284),(95,318,130,283),(96,317,131,282),(97,316,132,281),(98,315,133,350),(99,314,134,349),(100,313,135,348),(101,312,136,347),(102,311,137,346),(103,310,138,345),(104,309,139,344),(105,308,140,343),(141,375,176,410),(142,374,177,409),(143,373,178,408),(144,372,179,407),(145,371,180,406),(146,370,181,405),(147,369,182,404),(148,368,183,403),(149,367,184,402),(150,366,185,401),(151,365,186,400),(152,364,187,399),(153,363,188,398),(154,362,189,397),(155,361,190,396),(156,360,191,395),(157,359,192,394),(158,358,193,393),(159,357,194,392),(160,356,195,391),(161,355,196,390),(162,354,197,389),(163,353,198,388),(164,352,199,387),(165,351,200,386),(166,420,201,385),(167,419,202,384),(168,418,203,383),(169,417,204,382),(170,416,205,381),(171,415,206,380),(172,414,207,379),(173,413,208,378),(174,412,209,377),(175,411,210,376)]])

114 conjugacy classes

 class 1 2 3A 3B 4A 4B 5A 5B 6A 6B 7A 7B 7C 10A 10B 12A 12B 12C 12D 14A 14B 14C 15A 15B 15C 15D 21A ··· 21F 30A 30B 30C 30D 35A ··· 35L 42A ··· 42F 70A ··· 70L 105A ··· 105X 210A ··· 210X order 1 2 3 3 4 4 5 5 6 6 7 7 7 10 10 12 12 12 12 14 14 14 15 15 15 15 21 ··· 21 30 30 30 30 35 ··· 35 42 ··· 42 70 ··· 70 105 ··· 105 210 ··· 210 size 1 1 1 1 35 35 2 2 1 1 2 2 2 2 2 35 35 35 35 2 2 2 2 2 2 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

114 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - - + - image C1 C2 C3 C4 C6 C12 D5 D7 Dic5 Dic7 C3×D5 C3×D7 C3×Dic5 D35 C3×Dic7 Dic35 C3×D35 C3×Dic35 kernel C3×Dic35 C210 Dic35 C105 C70 C35 C42 C30 C21 C15 C14 C10 C7 C6 C5 C3 C2 C1 # reps 1 1 2 2 2 4 2 3 2 3 4 6 4 12 6 12 24 24

Matrix representation of C3×Dic35 in GL2(𝔽421) generated by

 400 0 0 400
,
 77 38 187 311
,
 97 420 148 324
G:=sub<GL(2,GF(421))| [400,0,0,400],[77,187,38,311],[97,148,420,324] >;

C3×Dic35 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{35}
% in TeX

G:=Group("C3xDic35");
// GroupNames label

G:=SmallGroup(420,7);
// by ID

G=gap.SmallGroup(420,7);
# by ID

G:=PCGroup([5,-2,-3,-2,-5,-7,30,963,9004]);
// Polycyclic

G:=Group<a,b,c|a^3=b^70=1,c^2=b^35,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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