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G = C3×Dic35order 420 = 22·3·5·7

Direct product of C3 and Dic35

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

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

C1C35 — C3×Dic35
C1C7C35C70C210 — C3×Dic35
C35 — C3×Dic35
C1C6

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

35C4
35C12
7Dic5
5Dic7
7C3×Dic5
5C3×Dic7

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

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

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

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

114 conjugacy classes

class 1  2 3A3B4A4B5A5B6A6B7A7B7C10A10B12A12B12C12D14A14B14C15A15B15C15D21A···21F30A30B30C30D35A···35L42A···42F70A···70L105A···105X210A···210X
order12334455667771010121212121414141515151521···213030303035···3542···4270···70105···105210···210
size111135352211222223535353522222222···222222···22···22···22···22···2

114 irreducible representations

dim111111222222222222
type++++--+-
imageC1C2C3C4C6C12D5D7Dic5Dic7C3×D5C3×D7C3×Dic5D35C3×Dic7Dic35C3×D35C3×Dic35
kernelC3×Dic35C210Dic35C105C70C35C42C30C21C15C14C10C7C6C5C3C2C1
# reps1122242323464126122424

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

4000
0400
,
7738
187311
,
97420
148324
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

Export

Subgroup lattice of C3×Dic35 in TeX

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