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G = C5×Dic21order 420 = 22·3·5·7

Direct product of C5 and Dic21

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

Aliases: C5×Dic21, C1057C4, C211C20, C70.2S3, C30.3D7, C354Dic3, C155Dic7, C42.1C10, C210.3C2, C10.2D21, C6.(C5×D7), C7⋊(C5×Dic3), C3⋊(C5×Dic7), C14.(C5×S3), C2.(C5×D21), SmallGroup(420,9)

Series: Derived Chief Lower central Upper central

C1C21 — C5×Dic21
C1C7C21C42C210 — C5×Dic21
C21 — C5×Dic21
C1C10

Generators and relations for C5×Dic21
 G = < a,b,c | a5=b42=1, c2=b21, ab=ba, ac=ca, cbc-1=b-1 >

21C4
7Dic3
21C20
3Dic7
7C5×Dic3
3C5×Dic7

Smallest permutation representation of C5×Dic21
Regular action on 420 points
Generators in S420
(1 206 137 100 58)(2 207 138 101 59)(3 208 139 102 60)(4 209 140 103 61)(5 210 141 104 62)(6 169 142 105 63)(7 170 143 106 64)(8 171 144 107 65)(9 172 145 108 66)(10 173 146 109 67)(11 174 147 110 68)(12 175 148 111 69)(13 176 149 112 70)(14 177 150 113 71)(15 178 151 114 72)(16 179 152 115 73)(17 180 153 116 74)(18 181 154 117 75)(19 182 155 118 76)(20 183 156 119 77)(21 184 157 120 78)(22 185 158 121 79)(23 186 159 122 80)(24 187 160 123 81)(25 188 161 124 82)(26 189 162 125 83)(27 190 163 126 84)(28 191 164 85 43)(29 192 165 86 44)(30 193 166 87 45)(31 194 167 88 46)(32 195 168 89 47)(33 196 127 90 48)(34 197 128 91 49)(35 198 129 92 50)(36 199 130 93 51)(37 200 131 94 52)(38 201 132 95 53)(39 202 133 96 54)(40 203 134 97 55)(41 204 135 98 56)(42 205 136 99 57)(211 386 365 323 288)(212 387 366 324 289)(213 388 367 325 290)(214 389 368 326 291)(215 390 369 327 292)(216 391 370 328 293)(217 392 371 329 294)(218 393 372 330 253)(219 394 373 331 254)(220 395 374 332 255)(221 396 375 333 256)(222 397 376 334 257)(223 398 377 335 258)(224 399 378 336 259)(225 400 337 295 260)(226 401 338 296 261)(227 402 339 297 262)(228 403 340 298 263)(229 404 341 299 264)(230 405 342 300 265)(231 406 343 301 266)(232 407 344 302 267)(233 408 345 303 268)(234 409 346 304 269)(235 410 347 305 270)(236 411 348 306 271)(237 412 349 307 272)(238 413 350 308 273)(239 414 351 309 274)(240 415 352 310 275)(241 416 353 311 276)(242 417 354 312 277)(243 418 355 313 278)(244 419 356 314 279)(245 420 357 315 280)(246 379 358 316 281)(247 380 359 317 282)(248 381 360 318 283)(249 382 361 319 284)(250 383 362 320 285)(251 384 363 321 286)(252 385 364 322 287)
(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 22 246)(2 224 23 245)(3 223 24 244)(4 222 25 243)(5 221 26 242)(6 220 27 241)(7 219 28 240)(8 218 29 239)(9 217 30 238)(10 216 31 237)(11 215 32 236)(12 214 33 235)(13 213 34 234)(14 212 35 233)(15 211 36 232)(16 252 37 231)(17 251 38 230)(18 250 39 229)(19 249 40 228)(20 248 41 227)(21 247 42 226)(43 275 64 254)(44 274 65 253)(45 273 66 294)(46 272 67 293)(47 271 68 292)(48 270 69 291)(49 269 70 290)(50 268 71 289)(51 267 72 288)(52 266 73 287)(53 265 74 286)(54 264 75 285)(55 263 76 284)(56 262 77 283)(57 261 78 282)(58 260 79 281)(59 259 80 280)(60 258 81 279)(61 257 82 278)(62 256 83 277)(63 255 84 276)(85 310 106 331)(86 309 107 330)(87 308 108 329)(88 307 109 328)(89 306 110 327)(90 305 111 326)(91 304 112 325)(92 303 113 324)(93 302 114 323)(94 301 115 322)(95 300 116 321)(96 299 117 320)(97 298 118 319)(98 297 119 318)(99 296 120 317)(100 295 121 316)(101 336 122 315)(102 335 123 314)(103 334 124 313)(104 333 125 312)(105 332 126 311)(127 347 148 368)(128 346 149 367)(129 345 150 366)(130 344 151 365)(131 343 152 364)(132 342 153 363)(133 341 154 362)(134 340 155 361)(135 339 156 360)(136 338 157 359)(137 337 158 358)(138 378 159 357)(139 377 160 356)(140 376 161 355)(141 375 162 354)(142 374 163 353)(143 373 164 352)(144 372 165 351)(145 371 166 350)(146 370 167 349)(147 369 168 348)(169 395 190 416)(170 394 191 415)(171 393 192 414)(172 392 193 413)(173 391 194 412)(174 390 195 411)(175 389 196 410)(176 388 197 409)(177 387 198 408)(178 386 199 407)(179 385 200 406)(180 384 201 405)(181 383 202 404)(182 382 203 403)(183 381 204 402)(184 380 205 401)(185 379 206 400)(186 420 207 399)(187 419 208 398)(188 418 209 397)(189 417 210 396)

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

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

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

120 conjugacy classes

class 1  2  3 4A4B5A5B5C5D 6 7A7B7C10A10B10C10D14A14B14C15A15B15C15D20A···20H21A···21F30A30B30C30D35A···35L42A···42F70A···70L105A···105X210A···210X
order1234455556777101010101414141515151520···2021···213030303035···3542···4270···70105···105210···210
size1122121111122221111222222221···212···222222···22···22···22···22···2

120 irreducible representations

dim111111222222222222
type+++-+-+-
imageC1C2C4C5C10C20S3Dic3D7Dic7C5×S3D21C5×Dic3C5×D7Dic21C5×Dic7C5×D21C5×Dic21
kernelC5×Dic21C210C105Dic21C42C21C70C35C30C15C14C10C7C6C5C3C2C1
# reps1124481133464126122424

Matrix representation of C5×Dic21 in GL2(𝔽41) generated by

100
010
,
2620
3327
,
94
032
G:=sub<GL(2,GF(41))| [10,0,0,10],[26,33,20,27],[9,0,4,32] >;

C5×Dic21 in GAP, Magma, Sage, TeX

C_5\times {\rm Dic}_{21}
% in TeX

G:=Group("C5xDic21");
// GroupNames label

G:=SmallGroup(420,9);
// by ID

G=gap.SmallGroup(420,9);
# by ID

G:=PCGroup([5,-2,-5,-2,-3,-7,50,803,9004]);
// Polycyclic

G:=Group<a,b,c|a^5=b^42=1,c^2=b^21,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×Dic21 in TeX

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