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## G = Dic5×C21order 420 = 22·3·5·7

### Direct product of C21 and Dic5

Aliases: Dic5×C21, C52C84, C154C28, C10.C42, C3511C12, C10510C4, C70.7C6, C42.4D5, C210.6C2, C30.2C14, C2.(D5×C21), C6.2(C7×D5), C14.4(C3×D5), SmallGroup(420,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — Dic5×C21
 Chief series C1 — C5 — C10 — C70 — C210 — Dic5×C21
 Lower central C5 — Dic5×C21
 Upper central C1 — C42

Generators and relations for Dic5×C21
G = < a,b,c | a21=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of Dic5×C21
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)
(1 178 341 208 359 320 77 155 53 379)(2 179 342 209 360 321 78 156 54 380)(3 180 343 210 361 322 79 157 55 381)(4 181 344 190 362 323 80 158 56 382)(5 182 345 191 363 324 81 159 57 383)(6 183 346 192 364 325 82 160 58 384)(7 184 347 193 365 326 83 161 59 385)(8 185 348 194 366 327 84 162 60 386)(9 186 349 195 367 328 64 163 61 387)(10 187 350 196 368 329 65 164 62 388)(11 188 351 197 369 330 66 165 63 389)(12 189 352 198 370 331 67 166 43 390)(13 169 353 199 371 332 68 167 44 391)(14 170 354 200 372 333 69 168 45 392)(15 171 355 201 373 334 70 148 46 393)(16 172 356 202 374 335 71 149 47 394)(17 173 357 203 375 336 72 150 48 395)(18 174 337 204 376 316 73 151 49 396)(19 175 338 205 377 317 74 152 50 397)(20 176 339 206 378 318 75 153 51 398)(21 177 340 207 358 319 76 154 52 399)(22 292 408 221 126 94 270 300 234 144)(23 293 409 222 106 95 271 301 235 145)(24 294 410 223 107 96 272 302 236 146)(25 274 411 224 108 97 273 303 237 147)(26 275 412 225 109 98 253 304 238 127)(27 276 413 226 110 99 254 305 239 128)(28 277 414 227 111 100 255 306 240 129)(29 278 415 228 112 101 256 307 241 130)(30 279 416 229 113 102 257 308 242 131)(31 280 417 230 114 103 258 309 243 132)(32 281 418 231 115 104 259 310 244 133)(33 282 419 211 116 105 260 311 245 134)(34 283 420 212 117 85 261 312 246 135)(35 284 400 213 118 86 262 313 247 136)(36 285 401 214 119 87 263 314 248 137)(37 286 402 215 120 88 264 315 249 138)(38 287 403 216 121 89 265 295 250 139)(39 288 404 217 122 90 266 296 251 140)(40 289 405 218 123 91 267 297 252 141)(41 290 406 219 124 92 268 298 232 142)(42 291 407 220 125 93 269 299 233 143)
(1 413 320 305)(2 414 321 306)(3 415 322 307)(4 416 323 308)(5 417 324 309)(6 418 325 310)(7 419 326 311)(8 420 327 312)(9 400 328 313)(10 401 329 314)(11 402 330 315)(12 403 331 295)(13 404 332 296)(14 405 333 297)(15 406 334 298)(16 407 335 299)(17 408 336 300)(18 409 316 301)(19 410 317 302)(20 411 318 303)(21 412 319 304)(22 150 94 357)(23 151 95 337)(24 152 96 338)(25 153 97 339)(26 154 98 340)(27 155 99 341)(28 156 100 342)(29 157 101 343)(30 158 102 344)(31 159 103 345)(32 160 104 346)(33 161 105 347)(34 162 85 348)(35 163 86 349)(36 164 87 350)(37 165 88 351)(38 166 89 352)(39 167 90 353)(40 168 91 354)(41 148 92 355)(42 149 93 356)(43 121 198 139)(44 122 199 140)(45 123 200 141)(46 124 201 142)(47 125 202 143)(48 126 203 144)(49 106 204 145)(50 107 205 146)(51 108 206 147)(52 109 207 127)(53 110 208 128)(54 111 209 129)(55 112 210 130)(56 113 190 131)(57 114 191 132)(58 115 192 133)(59 116 193 134)(60 117 194 135)(61 118 195 136)(62 119 196 137)(63 120 197 138)(64 262 186 284)(65 263 187 285)(66 264 188 286)(67 265 189 287)(68 266 169 288)(69 267 170 289)(70 268 171 290)(71 269 172 291)(72 270 173 292)(73 271 174 293)(74 272 175 294)(75 273 176 274)(76 253 177 275)(77 254 178 276)(78 255 179 277)(79 256 180 278)(80 257 181 279)(81 258 182 280)(82 259 183 281)(83 260 184 282)(84 261 185 283)(211 365 245 385)(212 366 246 386)(213 367 247 387)(214 368 248 388)(215 369 249 389)(216 370 250 390)(217 371 251 391)(218 372 252 392)(219 373 232 393)(220 374 233 394)(221 375 234 395)(222 376 235 396)(223 377 236 397)(224 378 237 398)(225 358 238 399)(226 359 239 379)(227 360 240 380)(228 361 241 381)(229 362 242 382)(230 363 243 383)(231 364 244 384)

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

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

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

168 conjugacy classes

 class 1 2 3A 3B 4A 4B 5A 5B 6A 6B 7A ··· 7F 10A 10B 12A 12B 12C 12D 14A ··· 14F 15A 15B 15C 15D 21A ··· 21L 28A ··· 28L 30A 30B 30C 30D 35A ··· 35L 42A ··· 42L 70A ··· 70L 84A ··· 84X 105A ··· 105X 210A ··· 210X order 1 2 3 3 4 4 5 5 6 6 7 ··· 7 10 10 12 12 12 12 14 ··· 14 15 15 15 15 21 ··· 21 28 ··· 28 30 30 30 30 35 ··· 35 42 ··· 42 70 ··· 70 84 ··· 84 105 ··· 105 210 ··· 210 size 1 1 1 1 5 5 2 2 1 1 1 ··· 1 2 2 5 5 5 5 1 ··· 1 2 2 2 2 1 ··· 1 5 ··· 5 2 2 2 2 2 ··· 2 1 ··· 1 2 ··· 2 5 ··· 5 2 ··· 2 2 ··· 2

168 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 C3 C4 C6 C7 C12 C14 C21 C28 C42 C84 D5 Dic5 C3×D5 C3×Dic5 C7×D5 C7×Dic5 D5×C21 Dic5×C21 kernel Dic5×C21 C210 C7×Dic5 C105 C70 C3×Dic5 C35 C30 Dic5 C15 C10 C5 C42 C21 C14 C7 C6 C3 C2 C1 # reps 1 1 2 2 2 6 4 6 12 12 12 24 2 2 4 4 12 12 24 24

Matrix representation of Dic5×C21 in GL3(𝔽421) generated by

 1 0 0 0 176 0 0 0 176
,
 420 0 0 0 0 1 0 420 310
,
 392 0 0 0 172 212 0 65 249
G:=sub<GL(3,GF(421))| [1,0,0,0,176,0,0,0,176],[420,0,0,0,0,420,0,1,310],[392,0,0,0,172,65,0,212,249] >;

Dic5×C21 in GAP, Magma, Sage, TeX

{\rm Dic}_5\times C_{21}
% in TeX

G:=Group("Dic5xC21");
// GroupNames label

G:=SmallGroup(420,6);
// by ID

G=gap.SmallGroup(420,6);
# by ID

G:=PCGroup([5,-2,-3,-7,-2,-5,210,8404]);
// Polycyclic

G:=Group<a,b,c|a^21=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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