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G = Dic5×C21order 420 = 22·3·5·7

Direct product of C21 and Dic5

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

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

C1C5 — Dic5×C21
C1C5C10C70C210 — Dic5×C21
C5 — Dic5×C21
C1C42

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

5C4
5C12
5C28
5C84

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

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

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

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

168 conjugacy classes

class 1  2 3A3B4A4B5A5B6A6B7A···7F10A10B12A12B12C12D14A···14F15A15B15C15D21A···21L28A···28L30A30B30C30D35A···35L42A···42L70A···70L84A···84X105A···105X210A···210X
order12334455667···710101212121214···141515151521···2128···283030303035···3542···4270···7084···84105···105210···210
size11115522111···12255551···122221···15···522222···21···12···25···52···22···2

168 irreducible representations

dim11111111111122222222
type+++-
imageC1C2C3C4C6C7C12C14C21C28C42C84D5Dic5C3×D5C3×Dic5C7×D5C7×Dic5D5×C21Dic5×C21
kernelDic5×C21C210C7×Dic5C105C70C3×Dic5C35C30Dic5C15C10C5C42C21C14C7C6C3C2C1
# reps1122264612121224224412122424

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

100
01760
00176
,
42000
001
0420310
,
39200
0172212
065249
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

Export

Subgroup lattice of Dic5×C21 in TeX

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