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G = Dic3×C35order 420 = 22·3·5·7

Direct product of C35 and Dic3

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

Aliases: Dic3×C35, C3⋊C140, C6.C70, C213C20, C155C28, C10511C4, 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

C1C3 — Dic3×C35
C1C3C6C42C210 — Dic3×C35
C3 — Dic3×C35
C1C70

Generators and relations for Dic3×C35
 G = < a,b,c | a35=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

3C4
3C20
3C28
3C140

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

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

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

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

210 conjugacy classes

class 1  2  3 4A4B5A5B5C5D 6 7A···7F10A10B10C10D14A···14F15A15B15C15D20A···20H21A···21F28A···28L30A30B30C30D35A···35X42A···42F70A···70X105A···105X140A···140AV210A···210X
order12344555567···71010101014···141515151520···2021···2128···283030303035···3542···4270···70105···105140···140210···210
size11233111121···111111···122223···32···23···322221···12···21···12···23···32···2

210 irreducible representations

dim11111111111122222222
type+++-
imageC1C2C4C5C7C10C14C20C28C35C70C140S3Dic3C5×S3S3×C7C5×Dic3C7×Dic3S3×C35Dic3×C35
kernelDic3×C35C210C105C7×Dic3C5×Dic3C42C30C21C15Dic3C6C3C70C35C14C10C7C5C2C1
# reps11246468122424481146462424

Matrix representation of Dic3×C35 in GL2(𝔽421) generated by

2910
0291
,
0420
11
,
4366
23378
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

Subgroup lattice of Dic3×C35 in TeX

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