direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C5×Dic24, C15⋊6Q32, C80.2S3, C48.1C10, C240.3C2, C40.77D6, C30.34D8, C20.41D12, C60.179D4, C10.15D24, C120.95C22, Dic12.1C10, C16.(C5×S3), C3⋊1(C5×Q32), C6.3(C5×D8), C2.5(C5×D24), C4.3(C5×D12), C8.15(S3×C10), C12.26(C5×D4), C24.16(C2×C10), (C5×Dic12).3C2, SmallGroup(480,120)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Dic24
G = < a,b,c | a5=b48=1, c2=b24, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×Dic24
►Regular action on 480 points
Generators in S480
(1 253 376 107 447)(2 254 377 108 448)(3 255 378 109 449)(4 256 379 110 450)(5 257 380 111 451)(6 258 381 112 452)(7 259 382 113 453)(8 260 383 114 454)(9 261 384 115 455)(10 262 337 116 456)(11 263 338 117 457)(12 264 339 118 458)(13 265 340 119 459)(14 266 341 120 460)(15 267 342 121 461)(16 268 343 122 462)(17 269 344 123 463)(18 270 345 124 464)(19 271 346 125 465)(20 272 347 126 466)(21 273 348 127 467)(22 274 349 128 468)(23 275 350 129 469)(24 276 351 130 470)(25 277 352 131 471)(26 278 353 132 472)(27 279 354 133 473)(28 280 355 134 474)(29 281 356 135 475)(30 282 357 136 476)(31 283 358 137 477)(32 284 359 138 478)(33 285 360 139 479)(34 286 361 140 480)(35 287 362 141 433)(36 288 363 142 434)(37 241 364 143 435)(38 242 365 144 436)(39 243 366 97 437)(40 244 367 98 438)(41 245 368 99 439)(42 246 369 100 440)(43 247 370 101 441)(44 248 371 102 442)(45 249 372 103 443)(46 250 373 104 444)(47 251 374 105 445)(48 252 375 106 446)(49 165 204 412 332)(50 166 205 413 333)(51 167 206 414 334)(52 168 207 415 335)(53 169 208 416 336)(54 170 209 417 289)(55 171 210 418 290)(56 172 211 419 291)(57 173 212 420 292)(58 174 213 421 293)(59 175 214 422 294)(60 176 215 423 295)(61 177 216 424 296)(62 178 217 425 297)(63 179 218 426 298)(64 180 219 427 299)(65 181 220 428 300)(66 182 221 429 301)(67 183 222 430 302)(68 184 223 431 303)(69 185 224 432 304)(70 186 225 385 305)(71 187 226 386 306)(72 188 227 387 307)(73 189 228 388 308)(74 190 229 389 309)(75 191 230 390 310)(76 192 231 391 311)(77 145 232 392 312)(78 146 233 393 313)(79 147 234 394 314)(80 148 235 395 315)(81 149 236 396 316)(82 150 237 397 317)(83 151 238 398 318)(84 152 239 399 319)(85 153 240 400 320)(86 154 193 401 321)(87 155 194 402 322)(88 156 195 403 323)(89 157 196 404 324)(90 158 197 405 325)(91 159 198 406 326)(92 160 199 407 327)(93 161 200 408 328)(94 162 201 409 329)(95 163 202 410 330)(96 164 203 411 331)
(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)(385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432)(433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480)
(1 405 25 429)(2 404 26 428)(3 403 27 427)(4 402 28 426)(5 401 29 425)(6 400 30 424)(7 399 31 423)(8 398 32 422)(9 397 33 421)(10 396 34 420)(11 395 35 419)(12 394 36 418)(13 393 37 417)(14 392 38 416)(15 391 39 415)(16 390 40 414)(17 389 41 413)(18 388 42 412)(19 387 43 411)(20 386 44 410)(21 385 45 409)(22 432 46 408)(23 431 47 407)(24 430 48 406)(49 345 73 369)(50 344 74 368)(51 343 75 367)(52 342 76 366)(53 341 77 365)(54 340 78 364)(55 339 79 363)(56 338 80 362)(57 337 81 361)(58 384 82 360)(59 383 83 359)(60 382 84 358)(61 381 85 357)(62 380 86 356)(63 379 87 355)(64 378 88 354)(65 377 89 353)(66 376 90 352)(67 375 91 351)(68 374 92 350)(69 373 93 349)(70 372 94 348)(71 371 95 347)(72 370 96 346)(97 168 121 192)(98 167 122 191)(99 166 123 190)(100 165 124 189)(101 164 125 188)(102 163 126 187)(103 162 127 186)(104 161 128 185)(105 160 129 184)(106 159 130 183)(107 158 131 182)(108 157 132 181)(109 156 133 180)(110 155 134 179)(111 154 135 178)(112 153 136 177)(113 152 137 176)(114 151 138 175)(115 150 139 174)(116 149 140 173)(117 148 141 172)(118 147 142 171)(119 146 143 170)(120 145 144 169)(193 475 217 451)(194 474 218 450)(195 473 219 449)(196 472 220 448)(197 471 221 447)(198 470 222 446)(199 469 223 445)(200 468 224 444)(201 467 225 443)(202 466 226 442)(203 465 227 441)(204 464 228 440)(205 463 229 439)(206 462 230 438)(207 461 231 437)(208 460 232 436)(209 459 233 435)(210 458 234 434)(211 457 235 433)(212 456 236 480)(213 455 237 479)(214 454 238 478)(215 453 239 477)(216 452 240 476)(241 289 265 313)(242 336 266 312)(243 335 267 311)(244 334 268 310)(245 333 269 309)(246 332 270 308)(247 331 271 307)(248 330 272 306)(249 329 273 305)(250 328 274 304)(251 327 275 303)(252 326 276 302)(253 325 277 301)(254 324 278 300)(255 323 279 299)(256 322 280 298)(257 321 281 297)(258 320 282 296)(259 319 283 295)(260 318 284 294)(261 317 285 293)(262 316 286 292)(263 315 287 291)(264 314 288 290)
G:=sub<Sym(480)| (1,253,376,107,447)(2,254,377,108,448)(3,255,378,109,449)(4,256,379,110,450)(5,257,380,111,451)(6,258,381,112,452)(7,259,382,113,453)(8,260,383,114,454)(9,261,384,115,455)(10,262,337,116,456)(11,263,338,117,457)(12,264,339,118,458)(13,265,340,119,459)(14,266,341,120,460)(15,267,342,121,461)(16,268,343,122,462)(17,269,344,123,463)(18,270,345,124,464)(19,271,346,125,465)(20,272,347,126,466)(21,273,348,127,467)(22,274,349,128,468)(23,275,350,129,469)(24,276,351,130,470)(25,277,352,131,471)(26,278,353,132,472)(27,279,354,133,473)(28,280,355,134,474)(29,281,356,135,475)(30,282,357,136,476)(31,283,358,137,477)(32,284,359,138,478)(33,285,360,139,479)(34,286,361,140,480)(35,287,362,141,433)(36,288,363,142,434)(37,241,364,143,435)(38,242,365,144,436)(39,243,366,97,437)(40,244,367,98,438)(41,245,368,99,439)(42,246,369,100,440)(43,247,370,101,441)(44,248,371,102,442)(45,249,372,103,443)(46,250,373,104,444)(47,251,374,105,445)(48,252,375,106,446)(49,165,204,412,332)(50,166,205,413,333)(51,167,206,414,334)(52,168,207,415,335)(53,169,208,416,336)(54,170,209,417,289)(55,171,210,418,290)(56,172,211,419,291)(57,173,212,420,292)(58,174,213,421,293)(59,175,214,422,294)(60,176,215,423,295)(61,177,216,424,296)(62,178,217,425,297)(63,179,218,426,298)(64,180,219,427,299)(65,181,220,428,300)(66,182,221,429,301)(67,183,222,430,302)(68,184,223,431,303)(69,185,224,432,304)(70,186,225,385,305)(71,187,226,386,306)(72,188,227,387,307)(73,189,228,388,308)(74,190,229,389,309)(75,191,230,390,310)(76,192,231,391,311)(77,145,232,392,312)(78,146,233,393,313)(79,147,234,394,314)(80,148,235,395,315)(81,149,236,396,316)(82,150,237,397,317)(83,151,238,398,318)(84,152,239,399,319)(85,153,240,400,320)(86,154,193,401,321)(87,155,194,402,322)(88,156,195,403,323)(89,157,196,404,324)(90,158,197,405,325)(91,159,198,406,326)(92,160,199,407,327)(93,161,200,408,328)(94,162,201,409,329)(95,163,202,410,330)(96,164,203,411,331), (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)(385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432)(433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480), (1,405,25,429)(2,404,26,428)(3,403,27,427)(4,402,28,426)(5,401,29,425)(6,400,30,424)(7,399,31,423)(8,398,32,422)(9,397,33,421)(10,396,34,420)(11,395,35,419)(12,394,36,418)(13,393,37,417)(14,392,38,416)(15,391,39,415)(16,390,40,414)(17,389,41,413)(18,388,42,412)(19,387,43,411)(20,386,44,410)(21,385,45,409)(22,432,46,408)(23,431,47,407)(24,430,48,406)(49,345,73,369)(50,344,74,368)(51,343,75,367)(52,342,76,366)(53,341,77,365)(54,340,78,364)(55,339,79,363)(56,338,80,362)(57,337,81,361)(58,384,82,360)(59,383,83,359)(60,382,84,358)(61,381,85,357)(62,380,86,356)(63,379,87,355)(64,378,88,354)(65,377,89,353)(66,376,90,352)(67,375,91,351)(68,374,92,350)(69,373,93,349)(70,372,94,348)(71,371,95,347)(72,370,96,346)(97,168,121,192)(98,167,122,191)(99,166,123,190)(100,165,124,189)(101,164,125,188)(102,163,126,187)(103,162,127,186)(104,161,128,185)(105,160,129,184)(106,159,130,183)(107,158,131,182)(108,157,132,181)(109,156,133,180)(110,155,134,179)(111,154,135,178)(112,153,136,177)(113,152,137,176)(114,151,138,175)(115,150,139,174)(116,149,140,173)(117,148,141,172)(118,147,142,171)(119,146,143,170)(120,145,144,169)(193,475,217,451)(194,474,218,450)(195,473,219,449)(196,472,220,448)(197,471,221,447)(198,470,222,446)(199,469,223,445)(200,468,224,444)(201,467,225,443)(202,466,226,442)(203,465,227,441)(204,464,228,440)(205,463,229,439)(206,462,230,438)(207,461,231,437)(208,460,232,436)(209,459,233,435)(210,458,234,434)(211,457,235,433)(212,456,236,480)(213,455,237,479)(214,454,238,478)(215,453,239,477)(216,452,240,476)(241,289,265,313)(242,336,266,312)(243,335,267,311)(244,334,268,310)(245,333,269,309)(246,332,270,308)(247,331,271,307)(248,330,272,306)(249,329,273,305)(250,328,274,304)(251,327,275,303)(252,326,276,302)(253,325,277,301)(254,324,278,300)(255,323,279,299)(256,322,280,298)(257,321,281,297)(258,320,282,296)(259,319,283,295)(260,318,284,294)(261,317,285,293)(262,316,286,292)(263,315,287,291)(264,314,288,290)>;
G:=Group( (1,253,376,107,447)(2,254,377,108,448)(3,255,378,109,449)(4,256,379,110,450)(5,257,380,111,451)(6,258,381,112,452)(7,259,382,113,453)(8,260,383,114,454)(9,261,384,115,455)(10,262,337,116,456)(11,263,338,117,457)(12,264,339,118,458)(13,265,340,119,459)(14,266,341,120,460)(15,267,342,121,461)(16,268,343,122,462)(17,269,344,123,463)(18,270,345,124,464)(19,271,346,125,465)(20,272,347,126,466)(21,273,348,127,467)(22,274,349,128,468)(23,275,350,129,469)(24,276,351,130,470)(25,277,352,131,471)(26,278,353,132,472)(27,279,354,133,473)(28,280,355,134,474)(29,281,356,135,475)(30,282,357,136,476)(31,283,358,137,477)(32,284,359,138,478)(33,285,360,139,479)(34,286,361,140,480)(35,287,362,141,433)(36,288,363,142,434)(37,241,364,143,435)(38,242,365,144,436)(39,243,366,97,437)(40,244,367,98,438)(41,245,368,99,439)(42,246,369,100,440)(43,247,370,101,441)(44,248,371,102,442)(45,249,372,103,443)(46,250,373,104,444)(47,251,374,105,445)(48,252,375,106,446)(49,165,204,412,332)(50,166,205,413,333)(51,167,206,414,334)(52,168,207,415,335)(53,169,208,416,336)(54,170,209,417,289)(55,171,210,418,290)(56,172,211,419,291)(57,173,212,420,292)(58,174,213,421,293)(59,175,214,422,294)(60,176,215,423,295)(61,177,216,424,296)(62,178,217,425,297)(63,179,218,426,298)(64,180,219,427,299)(65,181,220,428,300)(66,182,221,429,301)(67,183,222,430,302)(68,184,223,431,303)(69,185,224,432,304)(70,186,225,385,305)(71,187,226,386,306)(72,188,227,387,307)(73,189,228,388,308)(74,190,229,389,309)(75,191,230,390,310)(76,192,231,391,311)(77,145,232,392,312)(78,146,233,393,313)(79,147,234,394,314)(80,148,235,395,315)(81,149,236,396,316)(82,150,237,397,317)(83,151,238,398,318)(84,152,239,399,319)(85,153,240,400,320)(86,154,193,401,321)(87,155,194,402,322)(88,156,195,403,323)(89,157,196,404,324)(90,158,197,405,325)(91,159,198,406,326)(92,160,199,407,327)(93,161,200,408,328)(94,162,201,409,329)(95,163,202,410,330)(96,164,203,411,331), (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)(385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432)(433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480), (1,405,25,429)(2,404,26,428)(3,403,27,427)(4,402,28,426)(5,401,29,425)(6,400,30,424)(7,399,31,423)(8,398,32,422)(9,397,33,421)(10,396,34,420)(11,395,35,419)(12,394,36,418)(13,393,37,417)(14,392,38,416)(15,391,39,415)(16,390,40,414)(17,389,41,413)(18,388,42,412)(19,387,43,411)(20,386,44,410)(21,385,45,409)(22,432,46,408)(23,431,47,407)(24,430,48,406)(49,345,73,369)(50,344,74,368)(51,343,75,367)(52,342,76,366)(53,341,77,365)(54,340,78,364)(55,339,79,363)(56,338,80,362)(57,337,81,361)(58,384,82,360)(59,383,83,359)(60,382,84,358)(61,381,85,357)(62,380,86,356)(63,379,87,355)(64,378,88,354)(65,377,89,353)(66,376,90,352)(67,375,91,351)(68,374,92,350)(69,373,93,349)(70,372,94,348)(71,371,95,347)(72,370,96,346)(97,168,121,192)(98,167,122,191)(99,166,123,190)(100,165,124,189)(101,164,125,188)(102,163,126,187)(103,162,127,186)(104,161,128,185)(105,160,129,184)(106,159,130,183)(107,158,131,182)(108,157,132,181)(109,156,133,180)(110,155,134,179)(111,154,135,178)(112,153,136,177)(113,152,137,176)(114,151,138,175)(115,150,139,174)(116,149,140,173)(117,148,141,172)(118,147,142,171)(119,146,143,170)(120,145,144,169)(193,475,217,451)(194,474,218,450)(195,473,219,449)(196,472,220,448)(197,471,221,447)(198,470,222,446)(199,469,223,445)(200,468,224,444)(201,467,225,443)(202,466,226,442)(203,465,227,441)(204,464,228,440)(205,463,229,439)(206,462,230,438)(207,461,231,437)(208,460,232,436)(209,459,233,435)(210,458,234,434)(211,457,235,433)(212,456,236,480)(213,455,237,479)(214,454,238,478)(215,453,239,477)(216,452,240,476)(241,289,265,313)(242,336,266,312)(243,335,267,311)(244,334,268,310)(245,333,269,309)(246,332,270,308)(247,331,271,307)(248,330,272,306)(249,329,273,305)(250,328,274,304)(251,327,275,303)(252,326,276,302)(253,325,277,301)(254,324,278,300)(255,323,279,299)(256,322,280,298)(257,321,281,297)(258,320,282,296)(259,319,283,295)(260,318,284,294)(261,317,285,293)(262,316,286,292)(263,315,287,291)(264,314,288,290) );
G=PermutationGroup([[(1,253,376,107,447),(2,254,377,108,448),(3,255,378,109,449),(4,256,379,110,450),(5,257,380,111,451),(6,258,381,112,452),(7,259,382,113,453),(8,260,383,114,454),(9,261,384,115,455),(10,262,337,116,456),(11,263,338,117,457),(12,264,339,118,458),(13,265,340,119,459),(14,266,341,120,460),(15,267,342,121,461),(16,268,343,122,462),(17,269,344,123,463),(18,270,345,124,464),(19,271,346,125,465),(20,272,347,126,466),(21,273,348,127,467),(22,274,349,128,468),(23,275,350,129,469),(24,276,351,130,470),(25,277,352,131,471),(26,278,353,132,472),(27,279,354,133,473),(28,280,355,134,474),(29,281,356,135,475),(30,282,357,136,476),(31,283,358,137,477),(32,284,359,138,478),(33,285,360,139,479),(34,286,361,140,480),(35,287,362,141,433),(36,288,363,142,434),(37,241,364,143,435),(38,242,365,144,436),(39,243,366,97,437),(40,244,367,98,438),(41,245,368,99,439),(42,246,369,100,440),(43,247,370,101,441),(44,248,371,102,442),(45,249,372,103,443),(46,250,373,104,444),(47,251,374,105,445),(48,252,375,106,446),(49,165,204,412,332),(50,166,205,413,333),(51,167,206,414,334),(52,168,207,415,335),(53,169,208,416,336),(54,170,209,417,289),(55,171,210,418,290),(56,172,211,419,291),(57,173,212,420,292),(58,174,213,421,293),(59,175,214,422,294),(60,176,215,423,295),(61,177,216,424,296),(62,178,217,425,297),(63,179,218,426,298),(64,180,219,427,299),(65,181,220,428,300),(66,182,221,429,301),(67,183,222,430,302),(68,184,223,431,303),(69,185,224,432,304),(70,186,225,385,305),(71,187,226,386,306),(72,188,227,387,307),(73,189,228,388,308),(74,190,229,389,309),(75,191,230,390,310),(76,192,231,391,311),(77,145,232,392,312),(78,146,233,393,313),(79,147,234,394,314),(80,148,235,395,315),(81,149,236,396,316),(82,150,237,397,317),(83,151,238,398,318),(84,152,239,399,319),(85,153,240,400,320),(86,154,193,401,321),(87,155,194,402,322),(88,156,195,403,323),(89,157,196,404,324),(90,158,197,405,325),(91,159,198,406,326),(92,160,199,407,327),(93,161,200,408,328),(94,162,201,409,329),(95,163,202,410,330),(96,164,203,411,331)], [(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),(385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432),(433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480)], [(1,405,25,429),(2,404,26,428),(3,403,27,427),(4,402,28,426),(5,401,29,425),(6,400,30,424),(7,399,31,423),(8,398,32,422),(9,397,33,421),(10,396,34,420),(11,395,35,419),(12,394,36,418),(13,393,37,417),(14,392,38,416),(15,391,39,415),(16,390,40,414),(17,389,41,413),(18,388,42,412),(19,387,43,411),(20,386,44,410),(21,385,45,409),(22,432,46,408),(23,431,47,407),(24,430,48,406),(49,345,73,369),(50,344,74,368),(51,343,75,367),(52,342,76,366),(53,341,77,365),(54,340,78,364),(55,339,79,363),(56,338,80,362),(57,337,81,361),(58,384,82,360),(59,383,83,359),(60,382,84,358),(61,381,85,357),(62,380,86,356),(63,379,87,355),(64,378,88,354),(65,377,89,353),(66,376,90,352),(67,375,91,351),(68,374,92,350),(69,373,93,349),(70,372,94,348),(71,371,95,347),(72,370,96,346),(97,168,121,192),(98,167,122,191),(99,166,123,190),(100,165,124,189),(101,164,125,188),(102,163,126,187),(103,162,127,186),(104,161,128,185),(105,160,129,184),(106,159,130,183),(107,158,131,182),(108,157,132,181),(109,156,133,180),(110,155,134,179),(111,154,135,178),(112,153,136,177),(113,152,137,176),(114,151,138,175),(115,150,139,174),(116,149,140,173),(117,148,141,172),(118,147,142,171),(119,146,143,170),(120,145,144,169),(193,475,217,451),(194,474,218,450),(195,473,219,449),(196,472,220,448),(197,471,221,447),(198,470,222,446),(199,469,223,445),(200,468,224,444),(201,467,225,443),(202,466,226,442),(203,465,227,441),(204,464,228,440),(205,463,229,439),(206,462,230,438),(207,461,231,437),(208,460,232,436),(209,459,233,435),(210,458,234,434),(211,457,235,433),(212,456,236,480),(213,455,237,479),(214,454,238,478),(215,453,239,477),(216,452,240,476),(241,289,265,313),(242,336,266,312),(243,335,267,311),(244,334,268,310),(245,333,269,309),(246,332,270,308),(247,331,271,307),(248,330,272,306),(249,329,273,305),(250,328,274,304),(251,327,275,303),(252,326,276,302),(253,325,277,301),(254,324,278,300),(255,323,279,299),(256,322,280,298),(257,321,281,297),(258,320,282,296),(259,319,283,295),(260,318,284,294),(261,317,285,293),(262,316,286,292),(263,315,287,291),(264,314,288,290)]])
135 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6 | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 15A | 15B | 15C | 15D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 24A | 24B | 24C | 24D | 30A | 30B | 30C | 30D | 40A | ··· | 40H | 48A | ··· | 48H | 60A | ··· | 60H | 80A | ··· | 80P | 120A | ··· | 120P | 240A | ··· | 240AF |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | ··· | 60 | 80 | ··· | 80 | 120 | ··· | 120 | 240 | ··· | 240 |
| size | 1 | 1 | 2 | 2 | 24 | 24 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 24 | ··· | 24 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
135 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | + | - | |||||||||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | S3 | D4 | D6 | D8 | D12 | C5×S3 | Q32 | C5×D4 | D24 | S3×C10 | C5×D8 | Dic24 | C5×D12 | C5×Q32 | C5×D24 | C5×Dic24 |
| kernel | C5×Dic24 | C240 | C5×Dic12 | Dic24 | C48 | Dic12 | C80 | C60 | C40 | C30 | C20 | C16 | C15 | C12 | C10 | C8 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C5×Dic24 ►in GL4(𝔽241) generated by
| 205 | 0 | 0 | 0 |
| 0 | 205 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 96 | 66 | 0 | 0 |
| 137 | 167 | 0 | 0 |
| 0 | 0 | 44 | 178 |
| 0 | 0 | 63 | 107 |
| 205 | 29 | 0 | 0 |
| 80 | 36 | 0 | 0 |
| 0 | 0 | 116 | 207 |
| 0 | 0 | 91 | 125 |
G:=sub<GL(4,GF(241))| [205,0,0,0,0,205,0,0,0,0,1,0,0,0,0,1],[96,137,0,0,66,167,0,0,0,0,44,63,0,0,178,107],[205,80,0,0,29,36,0,0,0,0,116,91,0,0,207,125] >;
C5×Dic24 in GAP, Magma, Sage, TeX
C_5\times {\rm Dic}_{24} % in TeX
G:=Group("C5xDic24"); // GroupNames label
G:=SmallGroup(480,120);
// by ID
G=gap.SmallGroup(480,120);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,560,309,428,1683,192,4204,102,15686]);
// Polycyclic
G:=Group<a,b,c|a^5=b^48=1,c^2=b^24,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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