direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×Dic21, C105⋊7C4, C21⋊1C20, C70.2S3, C30.3D7, C35⋊4Dic3, C15⋊5Dic7, 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
| C21 — C5×Dic21 |
Generators and relations for C5×Dic21
G = < a,b,c | a5=b42=1, c2=b21, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×Dic21
►Regular action on 420 points
Generators in S420
(1 184 150 95 73)(2 185 151 96 74)(3 186 152 97 75)(4 187 153 98 76)(5 188 154 99 77)(6 189 155 100 78)(7 190 156 101 79)(8 191 157 102 80)(9 192 158 103 81)(10 193 159 104 82)(11 194 160 105 83)(12 195 161 106 84)(13 196 162 107 43)(14 197 163 108 44)(15 198 164 109 45)(16 199 165 110 46)(17 200 166 111 47)(18 201 167 112 48)(19 202 168 113 49)(20 203 127 114 50)(21 204 128 115 51)(22 205 129 116 52)(23 206 130 117 53)(24 207 131 118 54)(25 208 132 119 55)(26 209 133 120 56)(27 210 134 121 57)(28 169 135 122 58)(29 170 136 123 59)(30 171 137 124 60)(31 172 138 125 61)(32 173 139 126 62)(33 174 140 85 63)(34 175 141 86 64)(35 176 142 87 65)(36 177 143 88 66)(37 178 144 89 67)(38 179 145 90 68)(39 180 146 91 69)(40 181 147 92 70)(41 182 148 93 71)(42 183 149 94 72)(211 407 337 295 253)(212 408 338 296 254)(213 409 339 297 255)(214 410 340 298 256)(215 411 341 299 257)(216 412 342 300 258)(217 413 343 301 259)(218 414 344 302 260)(219 415 345 303 261)(220 416 346 304 262)(221 417 347 305 263)(222 418 348 306 264)(223 419 349 307 265)(224 420 350 308 266)(225 379 351 309 267)(226 380 352 310 268)(227 381 353 311 269)(228 382 354 312 270)(229 383 355 313 271)(230 384 356 314 272)(231 385 357 315 273)(232 386 358 316 274)(233 387 359 317 275)(234 388 360 318 276)(235 389 361 319 277)(236 390 362 320 278)(237 391 363 321 279)(238 392 364 322 280)(239 393 365 323 281)(240 394 366 324 282)(241 395 367 325 283)(242 396 368 326 284)(243 397 369 327 285)(244 398 370 328 286)(245 399 371 329 287)(246 400 372 330 288)(247 401 373 331 289)(248 402 374 332 290)(249 403 375 333 291)(250 404 376 334 292)(251 405 377 335 293)(252 406 378 336 294)
(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)
(1 211 22 232)(2 252 23 231)(3 251 24 230)(4 250 25 229)(5 249 26 228)(6 248 27 227)(7 247 28 226)(8 246 29 225)(9 245 30 224)(10 244 31 223)(11 243 32 222)(12 242 33 221)(13 241 34 220)(14 240 35 219)(15 239 36 218)(16 238 37 217)(17 237 38 216)(18 236 39 215)(19 235 40 214)(20 234 41 213)(21 233 42 212)(43 283 64 262)(44 282 65 261)(45 281 66 260)(46 280 67 259)(47 279 68 258)(48 278 69 257)(49 277 70 256)(50 276 71 255)(51 275 72 254)(52 274 73 253)(53 273 74 294)(54 272 75 293)(55 271 76 292)(56 270 77 291)(57 269 78 290)(58 268 79 289)(59 267 80 288)(60 266 81 287)(61 265 82 286)(62 264 83 285)(63 263 84 284)(85 305 106 326)(86 304 107 325)(87 303 108 324)(88 302 109 323)(89 301 110 322)(90 300 111 321)(91 299 112 320)(92 298 113 319)(93 297 114 318)(94 296 115 317)(95 295 116 316)(96 336 117 315)(97 335 118 314)(98 334 119 313)(99 333 120 312)(100 332 121 311)(101 331 122 310)(102 330 123 309)(103 329 124 308)(104 328 125 307)(105 327 126 306)(127 360 148 339)(128 359 149 338)(129 358 150 337)(130 357 151 378)(131 356 152 377)(132 355 153 376)(133 354 154 375)(134 353 155 374)(135 352 156 373)(136 351 157 372)(137 350 158 371)(138 349 159 370)(139 348 160 369)(140 347 161 368)(141 346 162 367)(142 345 163 366)(143 344 164 365)(144 343 165 364)(145 342 166 363)(146 341 167 362)(147 340 168 361)(169 380 190 401)(170 379 191 400)(171 420 192 399)(172 419 193 398)(173 418 194 397)(174 417 195 396)(175 416 196 395)(176 415 197 394)(177 414 198 393)(178 413 199 392)(179 412 200 391)(180 411 201 390)(181 410 202 389)(182 409 203 388)(183 408 204 387)(184 407 205 386)(185 406 206 385)(186 405 207 384)(187 404 208 383)(188 403 209 382)(189 402 210 381)
G:=sub<Sym(420)| (1,184,150,95,73)(2,185,151,96,74)(3,186,152,97,75)(4,187,153,98,76)(5,188,154,99,77)(6,189,155,100,78)(7,190,156,101,79)(8,191,157,102,80)(9,192,158,103,81)(10,193,159,104,82)(11,194,160,105,83)(12,195,161,106,84)(13,196,162,107,43)(14,197,163,108,44)(15,198,164,109,45)(16,199,165,110,46)(17,200,166,111,47)(18,201,167,112,48)(19,202,168,113,49)(20,203,127,114,50)(21,204,128,115,51)(22,205,129,116,52)(23,206,130,117,53)(24,207,131,118,54)(25,208,132,119,55)(26,209,133,120,56)(27,210,134,121,57)(28,169,135,122,58)(29,170,136,123,59)(30,171,137,124,60)(31,172,138,125,61)(32,173,139,126,62)(33,174,140,85,63)(34,175,141,86,64)(35,176,142,87,65)(36,177,143,88,66)(37,178,144,89,67)(38,179,145,90,68)(39,180,146,91,69)(40,181,147,92,70)(41,182,148,93,71)(42,183,149,94,72)(211,407,337,295,253)(212,408,338,296,254)(213,409,339,297,255)(214,410,340,298,256)(215,411,341,299,257)(216,412,342,300,258)(217,413,343,301,259)(218,414,344,302,260)(219,415,345,303,261)(220,416,346,304,262)(221,417,347,305,263)(222,418,348,306,264)(223,419,349,307,265)(224,420,350,308,266)(225,379,351,309,267)(226,380,352,310,268)(227,381,353,311,269)(228,382,354,312,270)(229,383,355,313,271)(230,384,356,314,272)(231,385,357,315,273)(232,386,358,316,274)(233,387,359,317,275)(234,388,360,318,276)(235,389,361,319,277)(236,390,362,320,278)(237,391,363,321,279)(238,392,364,322,280)(239,393,365,323,281)(240,394,366,324,282)(241,395,367,325,283)(242,396,368,326,284)(243,397,369,327,285)(244,398,370,328,286)(245,399,371,329,287)(246,400,372,330,288)(247,401,373,331,289)(248,402,374,332,290)(249,403,375,333,291)(250,404,376,334,292)(251,405,377,335,293)(252,406,378,336,294), (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), (1,211,22,232)(2,252,23,231)(3,251,24,230)(4,250,25,229)(5,249,26,228)(6,248,27,227)(7,247,28,226)(8,246,29,225)(9,245,30,224)(10,244,31,223)(11,243,32,222)(12,242,33,221)(13,241,34,220)(14,240,35,219)(15,239,36,218)(16,238,37,217)(17,237,38,216)(18,236,39,215)(19,235,40,214)(20,234,41,213)(21,233,42,212)(43,283,64,262)(44,282,65,261)(45,281,66,260)(46,280,67,259)(47,279,68,258)(48,278,69,257)(49,277,70,256)(50,276,71,255)(51,275,72,254)(52,274,73,253)(53,273,74,294)(54,272,75,293)(55,271,76,292)(56,270,77,291)(57,269,78,290)(58,268,79,289)(59,267,80,288)(60,266,81,287)(61,265,82,286)(62,264,83,285)(63,263,84,284)(85,305,106,326)(86,304,107,325)(87,303,108,324)(88,302,109,323)(89,301,110,322)(90,300,111,321)(91,299,112,320)(92,298,113,319)(93,297,114,318)(94,296,115,317)(95,295,116,316)(96,336,117,315)(97,335,118,314)(98,334,119,313)(99,333,120,312)(100,332,121,311)(101,331,122,310)(102,330,123,309)(103,329,124,308)(104,328,125,307)(105,327,126,306)(127,360,148,339)(128,359,149,338)(129,358,150,337)(130,357,151,378)(131,356,152,377)(132,355,153,376)(133,354,154,375)(134,353,155,374)(135,352,156,373)(136,351,157,372)(137,350,158,371)(138,349,159,370)(139,348,160,369)(140,347,161,368)(141,346,162,367)(142,345,163,366)(143,344,164,365)(144,343,165,364)(145,342,166,363)(146,341,167,362)(147,340,168,361)(169,380,190,401)(170,379,191,400)(171,420,192,399)(172,419,193,398)(173,418,194,397)(174,417,195,396)(175,416,196,395)(176,415,197,394)(177,414,198,393)(178,413,199,392)(179,412,200,391)(180,411,201,390)(181,410,202,389)(182,409,203,388)(183,408,204,387)(184,407,205,386)(185,406,206,385)(186,405,207,384)(187,404,208,383)(188,403,209,382)(189,402,210,381)>;
G:=Group( (1,184,150,95,73)(2,185,151,96,74)(3,186,152,97,75)(4,187,153,98,76)(5,188,154,99,77)(6,189,155,100,78)(7,190,156,101,79)(8,191,157,102,80)(9,192,158,103,81)(10,193,159,104,82)(11,194,160,105,83)(12,195,161,106,84)(13,196,162,107,43)(14,197,163,108,44)(15,198,164,109,45)(16,199,165,110,46)(17,200,166,111,47)(18,201,167,112,48)(19,202,168,113,49)(20,203,127,114,50)(21,204,128,115,51)(22,205,129,116,52)(23,206,130,117,53)(24,207,131,118,54)(25,208,132,119,55)(26,209,133,120,56)(27,210,134,121,57)(28,169,135,122,58)(29,170,136,123,59)(30,171,137,124,60)(31,172,138,125,61)(32,173,139,126,62)(33,174,140,85,63)(34,175,141,86,64)(35,176,142,87,65)(36,177,143,88,66)(37,178,144,89,67)(38,179,145,90,68)(39,180,146,91,69)(40,181,147,92,70)(41,182,148,93,71)(42,183,149,94,72)(211,407,337,295,253)(212,408,338,296,254)(213,409,339,297,255)(214,410,340,298,256)(215,411,341,299,257)(216,412,342,300,258)(217,413,343,301,259)(218,414,344,302,260)(219,415,345,303,261)(220,416,346,304,262)(221,417,347,305,263)(222,418,348,306,264)(223,419,349,307,265)(224,420,350,308,266)(225,379,351,309,267)(226,380,352,310,268)(227,381,353,311,269)(228,382,354,312,270)(229,383,355,313,271)(230,384,356,314,272)(231,385,357,315,273)(232,386,358,316,274)(233,387,359,317,275)(234,388,360,318,276)(235,389,361,319,277)(236,390,362,320,278)(237,391,363,321,279)(238,392,364,322,280)(239,393,365,323,281)(240,394,366,324,282)(241,395,367,325,283)(242,396,368,326,284)(243,397,369,327,285)(244,398,370,328,286)(245,399,371,329,287)(246,400,372,330,288)(247,401,373,331,289)(248,402,374,332,290)(249,403,375,333,291)(250,404,376,334,292)(251,405,377,335,293)(252,406,378,336,294), (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), (1,211,22,232)(2,252,23,231)(3,251,24,230)(4,250,25,229)(5,249,26,228)(6,248,27,227)(7,247,28,226)(8,246,29,225)(9,245,30,224)(10,244,31,223)(11,243,32,222)(12,242,33,221)(13,241,34,220)(14,240,35,219)(15,239,36,218)(16,238,37,217)(17,237,38,216)(18,236,39,215)(19,235,40,214)(20,234,41,213)(21,233,42,212)(43,283,64,262)(44,282,65,261)(45,281,66,260)(46,280,67,259)(47,279,68,258)(48,278,69,257)(49,277,70,256)(50,276,71,255)(51,275,72,254)(52,274,73,253)(53,273,74,294)(54,272,75,293)(55,271,76,292)(56,270,77,291)(57,269,78,290)(58,268,79,289)(59,267,80,288)(60,266,81,287)(61,265,82,286)(62,264,83,285)(63,263,84,284)(85,305,106,326)(86,304,107,325)(87,303,108,324)(88,302,109,323)(89,301,110,322)(90,300,111,321)(91,299,112,320)(92,298,113,319)(93,297,114,318)(94,296,115,317)(95,295,116,316)(96,336,117,315)(97,335,118,314)(98,334,119,313)(99,333,120,312)(100,332,121,311)(101,331,122,310)(102,330,123,309)(103,329,124,308)(104,328,125,307)(105,327,126,306)(127,360,148,339)(128,359,149,338)(129,358,150,337)(130,357,151,378)(131,356,152,377)(132,355,153,376)(133,354,154,375)(134,353,155,374)(135,352,156,373)(136,351,157,372)(137,350,158,371)(138,349,159,370)(139,348,160,369)(140,347,161,368)(141,346,162,367)(142,345,163,366)(143,344,164,365)(144,343,165,364)(145,342,166,363)(146,341,167,362)(147,340,168,361)(169,380,190,401)(170,379,191,400)(171,420,192,399)(172,419,193,398)(173,418,194,397)(174,417,195,396)(175,416,196,395)(176,415,197,394)(177,414,198,393)(178,413,199,392)(179,412,200,391)(180,411,201,390)(181,410,202,389)(182,409,203,388)(183,408,204,387)(184,407,205,386)(185,406,206,385)(186,405,207,384)(187,404,208,383)(188,403,209,382)(189,402,210,381) );
G=PermutationGroup([[(1,184,150,95,73),(2,185,151,96,74),(3,186,152,97,75),(4,187,153,98,76),(5,188,154,99,77),(6,189,155,100,78),(7,190,156,101,79),(8,191,157,102,80),(9,192,158,103,81),(10,193,159,104,82),(11,194,160,105,83),(12,195,161,106,84),(13,196,162,107,43),(14,197,163,108,44),(15,198,164,109,45),(16,199,165,110,46),(17,200,166,111,47),(18,201,167,112,48),(19,202,168,113,49),(20,203,127,114,50),(21,204,128,115,51),(22,205,129,116,52),(23,206,130,117,53),(24,207,131,118,54),(25,208,132,119,55),(26,209,133,120,56),(27,210,134,121,57),(28,169,135,122,58),(29,170,136,123,59),(30,171,137,124,60),(31,172,138,125,61),(32,173,139,126,62),(33,174,140,85,63),(34,175,141,86,64),(35,176,142,87,65),(36,177,143,88,66),(37,178,144,89,67),(38,179,145,90,68),(39,180,146,91,69),(40,181,147,92,70),(41,182,148,93,71),(42,183,149,94,72),(211,407,337,295,253),(212,408,338,296,254),(213,409,339,297,255),(214,410,340,298,256),(215,411,341,299,257),(216,412,342,300,258),(217,413,343,301,259),(218,414,344,302,260),(219,415,345,303,261),(220,416,346,304,262),(221,417,347,305,263),(222,418,348,306,264),(223,419,349,307,265),(224,420,350,308,266),(225,379,351,309,267),(226,380,352,310,268),(227,381,353,311,269),(228,382,354,312,270),(229,383,355,313,271),(230,384,356,314,272),(231,385,357,315,273),(232,386,358,316,274),(233,387,359,317,275),(234,388,360,318,276),(235,389,361,319,277),(236,390,362,320,278),(237,391,363,321,279),(238,392,364,322,280),(239,393,365,323,281),(240,394,366,324,282),(241,395,367,325,283),(242,396,368,326,284),(243,397,369,327,285),(244,398,370,328,286),(245,399,371,329,287),(246,400,372,330,288),(247,401,373,331,289),(248,402,374,332,290),(249,403,375,333,291),(250,404,376,334,292),(251,405,377,335,293),(252,406,378,336,294)], [(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)], [(1,211,22,232),(2,252,23,231),(3,251,24,230),(4,250,25,229),(5,249,26,228),(6,248,27,227),(7,247,28,226),(8,246,29,225),(9,245,30,224),(10,244,31,223),(11,243,32,222),(12,242,33,221),(13,241,34,220),(14,240,35,219),(15,239,36,218),(16,238,37,217),(17,237,38,216),(18,236,39,215),(19,235,40,214),(20,234,41,213),(21,233,42,212),(43,283,64,262),(44,282,65,261),(45,281,66,260),(46,280,67,259),(47,279,68,258),(48,278,69,257),(49,277,70,256),(50,276,71,255),(51,275,72,254),(52,274,73,253),(53,273,74,294),(54,272,75,293),(55,271,76,292),(56,270,77,291),(57,269,78,290),(58,268,79,289),(59,267,80,288),(60,266,81,287),(61,265,82,286),(62,264,83,285),(63,263,84,284),(85,305,106,326),(86,304,107,325),(87,303,108,324),(88,302,109,323),(89,301,110,322),(90,300,111,321),(91,299,112,320),(92,298,113,319),(93,297,114,318),(94,296,115,317),(95,295,116,316),(96,336,117,315),(97,335,118,314),(98,334,119,313),(99,333,120,312),(100,332,121,311),(101,331,122,310),(102,330,123,309),(103,329,124,308),(104,328,125,307),(105,327,126,306),(127,360,148,339),(128,359,149,338),(129,358,150,337),(130,357,151,378),(131,356,152,377),(132,355,153,376),(133,354,154,375),(134,353,155,374),(135,352,156,373),(136,351,157,372),(137,350,158,371),(138,349,159,370),(139,348,160,369),(140,347,161,368),(141,346,162,367),(142,345,163,366),(143,344,164,365),(144,343,165,364),(145,342,166,363),(146,341,167,362),(147,340,168,361),(169,380,190,401),(170,379,191,400),(171,420,192,399),(172,419,193,398),(173,418,194,397),(174,417,195,396),(175,416,196,395),(176,415,197,394),(177,414,198,393),(178,413,199,392),(179,412,200,391),(180,411,201,390),(181,410,202,389),(182,409,203,388),(183,408,204,387),(184,407,205,386),(185,406,206,385),(186,405,207,384),(187,404,208,383),(188,403,209,382),(189,402,210,381)]])
120 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 5C | 5D | 6 | 7A | 7B | 7C | 10A | 10B | 10C | 10D | 14A | 14B | 14C | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 21A | ··· | 21F | 30A | 30B | 30C | 30D | 35A | ··· | 35L | 42A | ··· | 42F | 70A | ··· | 70L | 105A | ··· | 105X | 210A | ··· | 210X |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 7 | 7 | 7 | 10 | 10 | 10 | 10 | 14 | 14 | 14 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 21 | ··· | 21 | 30 | 30 | 30 | 30 | 35 | ··· | 35 | 42 | ··· | 42 | 70 | ··· | 70 | 105 | ··· | 105 | 210 | ··· | 210 |
| size | 1 | 1 | 2 | 21 | 21 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 21 | ··· | 21 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | ||||||||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | S3 | Dic3 | D7 | Dic7 | C5×S3 | D21 | C5×Dic3 | C5×D7 | Dic21 | C5×Dic7 | C5×D21 | C5×Dic21 |
| kernel | C5×Dic21 | C210 | C105 | Dic21 | C42 | C21 | C70 | C35 | C30 | C15 | C14 | C10 | C7 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 1 | 3 | 3 | 4 | 6 | 4 | 12 | 6 | 12 | 24 | 24 |
Matrix representation of C5×Dic21 ►in GL2(𝔽41) generated by
| 10 | 0 |
| 0 | 10 |
| 26 | 20 |
| 33 | 27 |
| 9 | 4 |
| 0 | 32 |
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