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## G = C5×C24⋊1C4order 480 = 25·3·5

### Direct product of C5 and C24⋊1C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×C24⋊1C4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C60 — C5×C4⋊Dic3 — C5×C24⋊1C4
 Lower central C3 — C6 — C12 — C5×C24⋊1C4
 Upper central C1 — C2×C10 — C2×C20 — C2×C40

Generators and relations for C5×C241C4
G = < a,b,c | a5=b24=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 164 in 72 conjugacy classes, 50 normal (38 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C8, C2×C4, C2×C4, C10, Dic3, C12, C2×C6, C15, C4⋊C4, C2×C8, C20, C20, C2×C10, C24, C2×Dic3, C2×C12, C30, C2.D8, C40, C2×C20, C2×C20, C4⋊Dic3, C2×C24, C5×Dic3, C60, C2×C30, C5×C4⋊C4, C2×C40, C241C4, C120, C10×Dic3, C2×C60, C5×C2.D8, C5×C4⋊Dic3, C2×C120, C5×C241C4
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, Q8, C10, Dic3, D6, C4⋊C4, D8, Q16, C20, C2×C10, Dic6, D12, C2×Dic3, C5×S3, C2.D8, C2×C20, C5×D4, C5×Q8, D24, Dic12, C4⋊Dic3, C5×Dic3, S3×C10, C5×C4⋊C4, C5×D8, C5×Q16, C241C4, C5×Dic6, C5×D12, C10×Dic3, C5×C2.D8, C5×D24, C5×Dic12, C5×C4⋊Dic3, C5×C241C4

Smallest permutation representation of C5×C241C4
Regular action on 480 points
Generators in S480
(1 199 262 144 148)(2 200 263 121 149)(3 201 264 122 150)(4 202 241 123 151)(5 203 242 124 152)(6 204 243 125 153)(7 205 244 126 154)(8 206 245 127 155)(9 207 246 128 156)(10 208 247 129 157)(11 209 248 130 158)(12 210 249 131 159)(13 211 250 132 160)(14 212 251 133 161)(15 213 252 134 162)(16 214 253 135 163)(17 215 254 136 164)(18 216 255 137 165)(19 193 256 138 166)(20 194 257 139 167)(21 195 258 140 168)(22 196 259 141 145)(23 197 260 142 146)(24 198 261 143 147)(25 358 288 53 323)(26 359 265 54 324)(27 360 266 55 325)(28 337 267 56 326)(29 338 268 57 327)(30 339 269 58 328)(31 340 270 59 329)(32 341 271 60 330)(33 342 272 61 331)(34 343 273 62 332)(35 344 274 63 333)(36 345 275 64 334)(37 346 276 65 335)(38 347 277 66 336)(39 348 278 67 313)(40 349 279 68 314)(41 350 280 69 315)(42 351 281 70 316)(43 352 282 71 317)(44 353 283 72 318)(45 354 284 49 319)(46 355 285 50 320)(47 356 286 51 321)(48 357 287 52 322)(73 400 374 430 235)(74 401 375 431 236)(75 402 376 432 237)(76 403 377 409 238)(77 404 378 410 239)(78 405 379 411 240)(79 406 380 412 217)(80 407 381 413 218)(81 408 382 414 219)(82 385 383 415 220)(83 386 384 416 221)(84 387 361 417 222)(85 388 362 418 223)(86 389 363 419 224)(87 390 364 420 225)(88 391 365 421 226)(89 392 366 422 227)(90 393 367 423 228)(91 394 368 424 229)(92 395 369 425 230)(93 396 370 426 231)(94 397 371 427 232)(95 398 372 428 233)(96 399 373 429 234)(97 473 170 292 447)(98 474 171 293 448)(99 475 172 294 449)(100 476 173 295 450)(101 477 174 296 451)(102 478 175 297 452)(103 479 176 298 453)(104 480 177 299 454)(105 457 178 300 455)(106 458 179 301 456)(107 459 180 302 433)(108 460 181 303 434)(109 461 182 304 435)(110 462 183 305 436)(111 463 184 306 437)(112 464 185 307 438)(113 465 186 308 439)(114 466 187 309 440)(115 467 188 310 441)(116 468 189 311 442)(117 469 190 312 443)(118 470 191 289 444)(119 471 192 290 445)(120 472 169 291 446)
(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)
(1 470 236 323)(2 469 237 322)(3 468 238 321)(4 467 239 320)(5 466 240 319)(6 465 217 318)(7 464 218 317)(8 463 219 316)(9 462 220 315)(10 461 221 314)(11 460 222 313)(12 459 223 336)(13 458 224 335)(14 457 225 334)(15 480 226 333)(16 479 227 332)(17 478 228 331)(18 477 229 330)(19 476 230 329)(20 475 231 328)(21 474 232 327)(22 473 233 326)(23 472 234 325)(24 471 235 324)(25 199 191 74)(26 198 192 73)(27 197 169 96)(28 196 170 95)(29 195 171 94)(30 194 172 93)(31 193 173 92)(32 216 174 91)(33 215 175 90)(34 214 176 89)(35 213 177 88)(36 212 178 87)(37 211 179 86)(38 210 180 85)(39 209 181 84)(40 208 182 83)(41 207 183 82)(42 206 184 81)(43 205 185 80)(44 204 186 79)(45 203 187 78)(46 202 188 77)(47 201 189 76)(48 200 190 75)(49 152 114 411)(50 151 115 410)(51 150 116 409)(52 149 117 432)(53 148 118 431)(54 147 119 430)(55 146 120 429)(56 145 97 428)(57 168 98 427)(58 167 99 426)(59 166 100 425)(60 165 101 424)(61 164 102 423)(62 163 103 422)(63 162 104 421)(64 161 105 420)(65 160 106 419)(66 159 107 418)(67 158 108 417)(68 157 109 416)(69 156 110 415)(70 155 111 414)(71 154 112 413)(72 153 113 412)(121 443 376 287)(122 442 377 286)(123 441 378 285)(124 440 379 284)(125 439 380 283)(126 438 381 282)(127 437 382 281)(128 436 383 280)(129 435 384 279)(130 434 361 278)(131 433 362 277)(132 456 363 276)(133 455 364 275)(134 454 365 274)(135 453 366 273)(136 452 367 272)(137 451 368 271)(138 450 369 270)(139 449 370 269)(140 448 371 268)(141 447 372 267)(142 446 373 266)(143 445 374 265)(144 444 375 288)(241 310 404 355)(242 309 405 354)(243 308 406 353)(244 307 407 352)(245 306 408 351)(246 305 385 350)(247 304 386 349)(248 303 387 348)(249 302 388 347)(250 301 389 346)(251 300 390 345)(252 299 391 344)(253 298 392 343)(254 297 393 342)(255 296 394 341)(256 295 395 340)(257 294 396 339)(258 293 397 338)(259 292 398 337)(260 291 399 360)(261 290 400 359)(262 289 401 358)(263 312 402 357)(264 311 403 356)

G:=sub<Sym(480)| (1,199,262,144,148)(2,200,263,121,149)(3,201,264,122,150)(4,202,241,123,151)(5,203,242,124,152)(6,204,243,125,153)(7,205,244,126,154)(8,206,245,127,155)(9,207,246,128,156)(10,208,247,129,157)(11,209,248,130,158)(12,210,249,131,159)(13,211,250,132,160)(14,212,251,133,161)(15,213,252,134,162)(16,214,253,135,163)(17,215,254,136,164)(18,216,255,137,165)(19,193,256,138,166)(20,194,257,139,167)(21,195,258,140,168)(22,196,259,141,145)(23,197,260,142,146)(24,198,261,143,147)(25,358,288,53,323)(26,359,265,54,324)(27,360,266,55,325)(28,337,267,56,326)(29,338,268,57,327)(30,339,269,58,328)(31,340,270,59,329)(32,341,271,60,330)(33,342,272,61,331)(34,343,273,62,332)(35,344,274,63,333)(36,345,275,64,334)(37,346,276,65,335)(38,347,277,66,336)(39,348,278,67,313)(40,349,279,68,314)(41,350,280,69,315)(42,351,281,70,316)(43,352,282,71,317)(44,353,283,72,318)(45,354,284,49,319)(46,355,285,50,320)(47,356,286,51,321)(48,357,287,52,322)(73,400,374,430,235)(74,401,375,431,236)(75,402,376,432,237)(76,403,377,409,238)(77,404,378,410,239)(78,405,379,411,240)(79,406,380,412,217)(80,407,381,413,218)(81,408,382,414,219)(82,385,383,415,220)(83,386,384,416,221)(84,387,361,417,222)(85,388,362,418,223)(86,389,363,419,224)(87,390,364,420,225)(88,391,365,421,226)(89,392,366,422,227)(90,393,367,423,228)(91,394,368,424,229)(92,395,369,425,230)(93,396,370,426,231)(94,397,371,427,232)(95,398,372,428,233)(96,399,373,429,234)(97,473,170,292,447)(98,474,171,293,448)(99,475,172,294,449)(100,476,173,295,450)(101,477,174,296,451)(102,478,175,297,452)(103,479,176,298,453)(104,480,177,299,454)(105,457,178,300,455)(106,458,179,301,456)(107,459,180,302,433)(108,460,181,303,434)(109,461,182,304,435)(110,462,183,305,436)(111,463,184,306,437)(112,464,185,307,438)(113,465,186,308,439)(114,466,187,309,440)(115,467,188,310,441)(116,468,189,311,442)(117,469,190,312,443)(118,470,191,289,444)(119,471,192,290,445)(120,472,169,291,446), (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), (1,470,236,323)(2,469,237,322)(3,468,238,321)(4,467,239,320)(5,466,240,319)(6,465,217,318)(7,464,218,317)(8,463,219,316)(9,462,220,315)(10,461,221,314)(11,460,222,313)(12,459,223,336)(13,458,224,335)(14,457,225,334)(15,480,226,333)(16,479,227,332)(17,478,228,331)(18,477,229,330)(19,476,230,329)(20,475,231,328)(21,474,232,327)(22,473,233,326)(23,472,234,325)(24,471,235,324)(25,199,191,74)(26,198,192,73)(27,197,169,96)(28,196,170,95)(29,195,171,94)(30,194,172,93)(31,193,173,92)(32,216,174,91)(33,215,175,90)(34,214,176,89)(35,213,177,88)(36,212,178,87)(37,211,179,86)(38,210,180,85)(39,209,181,84)(40,208,182,83)(41,207,183,82)(42,206,184,81)(43,205,185,80)(44,204,186,79)(45,203,187,78)(46,202,188,77)(47,201,189,76)(48,200,190,75)(49,152,114,411)(50,151,115,410)(51,150,116,409)(52,149,117,432)(53,148,118,431)(54,147,119,430)(55,146,120,429)(56,145,97,428)(57,168,98,427)(58,167,99,426)(59,166,100,425)(60,165,101,424)(61,164,102,423)(62,163,103,422)(63,162,104,421)(64,161,105,420)(65,160,106,419)(66,159,107,418)(67,158,108,417)(68,157,109,416)(69,156,110,415)(70,155,111,414)(71,154,112,413)(72,153,113,412)(121,443,376,287)(122,442,377,286)(123,441,378,285)(124,440,379,284)(125,439,380,283)(126,438,381,282)(127,437,382,281)(128,436,383,280)(129,435,384,279)(130,434,361,278)(131,433,362,277)(132,456,363,276)(133,455,364,275)(134,454,365,274)(135,453,366,273)(136,452,367,272)(137,451,368,271)(138,450,369,270)(139,449,370,269)(140,448,371,268)(141,447,372,267)(142,446,373,266)(143,445,374,265)(144,444,375,288)(241,310,404,355)(242,309,405,354)(243,308,406,353)(244,307,407,352)(245,306,408,351)(246,305,385,350)(247,304,386,349)(248,303,387,348)(249,302,388,347)(250,301,389,346)(251,300,390,345)(252,299,391,344)(253,298,392,343)(254,297,393,342)(255,296,394,341)(256,295,395,340)(257,294,396,339)(258,293,397,338)(259,292,398,337)(260,291,399,360)(261,290,400,359)(262,289,401,358)(263,312,402,357)(264,311,403,356)>;

G:=Group( (1,199,262,144,148)(2,200,263,121,149)(3,201,264,122,150)(4,202,241,123,151)(5,203,242,124,152)(6,204,243,125,153)(7,205,244,126,154)(8,206,245,127,155)(9,207,246,128,156)(10,208,247,129,157)(11,209,248,130,158)(12,210,249,131,159)(13,211,250,132,160)(14,212,251,133,161)(15,213,252,134,162)(16,214,253,135,163)(17,215,254,136,164)(18,216,255,137,165)(19,193,256,138,166)(20,194,257,139,167)(21,195,258,140,168)(22,196,259,141,145)(23,197,260,142,146)(24,198,261,143,147)(25,358,288,53,323)(26,359,265,54,324)(27,360,266,55,325)(28,337,267,56,326)(29,338,268,57,327)(30,339,269,58,328)(31,340,270,59,329)(32,341,271,60,330)(33,342,272,61,331)(34,343,273,62,332)(35,344,274,63,333)(36,345,275,64,334)(37,346,276,65,335)(38,347,277,66,336)(39,348,278,67,313)(40,349,279,68,314)(41,350,280,69,315)(42,351,281,70,316)(43,352,282,71,317)(44,353,283,72,318)(45,354,284,49,319)(46,355,285,50,320)(47,356,286,51,321)(48,357,287,52,322)(73,400,374,430,235)(74,401,375,431,236)(75,402,376,432,237)(76,403,377,409,238)(77,404,378,410,239)(78,405,379,411,240)(79,406,380,412,217)(80,407,381,413,218)(81,408,382,414,219)(82,385,383,415,220)(83,386,384,416,221)(84,387,361,417,222)(85,388,362,418,223)(86,389,363,419,224)(87,390,364,420,225)(88,391,365,421,226)(89,392,366,422,227)(90,393,367,423,228)(91,394,368,424,229)(92,395,369,425,230)(93,396,370,426,231)(94,397,371,427,232)(95,398,372,428,233)(96,399,373,429,234)(97,473,170,292,447)(98,474,171,293,448)(99,475,172,294,449)(100,476,173,295,450)(101,477,174,296,451)(102,478,175,297,452)(103,479,176,298,453)(104,480,177,299,454)(105,457,178,300,455)(106,458,179,301,456)(107,459,180,302,433)(108,460,181,303,434)(109,461,182,304,435)(110,462,183,305,436)(111,463,184,306,437)(112,464,185,307,438)(113,465,186,308,439)(114,466,187,309,440)(115,467,188,310,441)(116,468,189,311,442)(117,469,190,312,443)(118,470,191,289,444)(119,471,192,290,445)(120,472,169,291,446), (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), (1,470,236,323)(2,469,237,322)(3,468,238,321)(4,467,239,320)(5,466,240,319)(6,465,217,318)(7,464,218,317)(8,463,219,316)(9,462,220,315)(10,461,221,314)(11,460,222,313)(12,459,223,336)(13,458,224,335)(14,457,225,334)(15,480,226,333)(16,479,227,332)(17,478,228,331)(18,477,229,330)(19,476,230,329)(20,475,231,328)(21,474,232,327)(22,473,233,326)(23,472,234,325)(24,471,235,324)(25,199,191,74)(26,198,192,73)(27,197,169,96)(28,196,170,95)(29,195,171,94)(30,194,172,93)(31,193,173,92)(32,216,174,91)(33,215,175,90)(34,214,176,89)(35,213,177,88)(36,212,178,87)(37,211,179,86)(38,210,180,85)(39,209,181,84)(40,208,182,83)(41,207,183,82)(42,206,184,81)(43,205,185,80)(44,204,186,79)(45,203,187,78)(46,202,188,77)(47,201,189,76)(48,200,190,75)(49,152,114,411)(50,151,115,410)(51,150,116,409)(52,149,117,432)(53,148,118,431)(54,147,119,430)(55,146,120,429)(56,145,97,428)(57,168,98,427)(58,167,99,426)(59,166,100,425)(60,165,101,424)(61,164,102,423)(62,163,103,422)(63,162,104,421)(64,161,105,420)(65,160,106,419)(66,159,107,418)(67,158,108,417)(68,157,109,416)(69,156,110,415)(70,155,111,414)(71,154,112,413)(72,153,113,412)(121,443,376,287)(122,442,377,286)(123,441,378,285)(124,440,379,284)(125,439,380,283)(126,438,381,282)(127,437,382,281)(128,436,383,280)(129,435,384,279)(130,434,361,278)(131,433,362,277)(132,456,363,276)(133,455,364,275)(134,454,365,274)(135,453,366,273)(136,452,367,272)(137,451,368,271)(138,450,369,270)(139,449,370,269)(140,448,371,268)(141,447,372,267)(142,446,373,266)(143,445,374,265)(144,444,375,288)(241,310,404,355)(242,309,405,354)(243,308,406,353)(244,307,407,352)(245,306,408,351)(246,305,385,350)(247,304,386,349)(248,303,387,348)(249,302,388,347)(250,301,389,346)(251,300,390,345)(252,299,391,344)(253,298,392,343)(254,297,393,342)(255,296,394,341)(256,295,395,340)(257,294,396,339)(258,293,397,338)(259,292,398,337)(260,291,399,360)(261,290,400,359)(262,289,401,358)(263,312,402,357)(264,311,403,356) );

G=PermutationGroup([[(1,199,262,144,148),(2,200,263,121,149),(3,201,264,122,150),(4,202,241,123,151),(5,203,242,124,152),(6,204,243,125,153),(7,205,244,126,154),(8,206,245,127,155),(9,207,246,128,156),(10,208,247,129,157),(11,209,248,130,158),(12,210,249,131,159),(13,211,250,132,160),(14,212,251,133,161),(15,213,252,134,162),(16,214,253,135,163),(17,215,254,136,164),(18,216,255,137,165),(19,193,256,138,166),(20,194,257,139,167),(21,195,258,140,168),(22,196,259,141,145),(23,197,260,142,146),(24,198,261,143,147),(25,358,288,53,323),(26,359,265,54,324),(27,360,266,55,325),(28,337,267,56,326),(29,338,268,57,327),(30,339,269,58,328),(31,340,270,59,329),(32,341,271,60,330),(33,342,272,61,331),(34,343,273,62,332),(35,344,274,63,333),(36,345,275,64,334),(37,346,276,65,335),(38,347,277,66,336),(39,348,278,67,313),(40,349,279,68,314),(41,350,280,69,315),(42,351,281,70,316),(43,352,282,71,317),(44,353,283,72,318),(45,354,284,49,319),(46,355,285,50,320),(47,356,286,51,321),(48,357,287,52,322),(73,400,374,430,235),(74,401,375,431,236),(75,402,376,432,237),(76,403,377,409,238),(77,404,378,410,239),(78,405,379,411,240),(79,406,380,412,217),(80,407,381,413,218),(81,408,382,414,219),(82,385,383,415,220),(83,386,384,416,221),(84,387,361,417,222),(85,388,362,418,223),(86,389,363,419,224),(87,390,364,420,225),(88,391,365,421,226),(89,392,366,422,227),(90,393,367,423,228),(91,394,368,424,229),(92,395,369,425,230),(93,396,370,426,231),(94,397,371,427,232),(95,398,372,428,233),(96,399,373,429,234),(97,473,170,292,447),(98,474,171,293,448),(99,475,172,294,449),(100,476,173,295,450),(101,477,174,296,451),(102,478,175,297,452),(103,479,176,298,453),(104,480,177,299,454),(105,457,178,300,455),(106,458,179,301,456),(107,459,180,302,433),(108,460,181,303,434),(109,461,182,304,435),(110,462,183,305,436),(111,463,184,306,437),(112,464,185,307,438),(113,465,186,308,439),(114,466,187,309,440),(115,467,188,310,441),(116,468,189,311,442),(117,469,190,312,443),(118,470,191,289,444),(119,471,192,290,445),(120,472,169,291,446)], [(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)], [(1,470,236,323),(2,469,237,322),(3,468,238,321),(4,467,239,320),(5,466,240,319),(6,465,217,318),(7,464,218,317),(8,463,219,316),(9,462,220,315),(10,461,221,314),(11,460,222,313),(12,459,223,336),(13,458,224,335),(14,457,225,334),(15,480,226,333),(16,479,227,332),(17,478,228,331),(18,477,229,330),(19,476,230,329),(20,475,231,328),(21,474,232,327),(22,473,233,326),(23,472,234,325),(24,471,235,324),(25,199,191,74),(26,198,192,73),(27,197,169,96),(28,196,170,95),(29,195,171,94),(30,194,172,93),(31,193,173,92),(32,216,174,91),(33,215,175,90),(34,214,176,89),(35,213,177,88),(36,212,178,87),(37,211,179,86),(38,210,180,85),(39,209,181,84),(40,208,182,83),(41,207,183,82),(42,206,184,81),(43,205,185,80),(44,204,186,79),(45,203,187,78),(46,202,188,77),(47,201,189,76),(48,200,190,75),(49,152,114,411),(50,151,115,410),(51,150,116,409),(52,149,117,432),(53,148,118,431),(54,147,119,430),(55,146,120,429),(56,145,97,428),(57,168,98,427),(58,167,99,426),(59,166,100,425),(60,165,101,424),(61,164,102,423),(62,163,103,422),(63,162,104,421),(64,161,105,420),(65,160,106,419),(66,159,107,418),(67,158,108,417),(68,157,109,416),(69,156,110,415),(70,155,111,414),(71,154,112,413),(72,153,113,412),(121,443,376,287),(122,442,377,286),(123,441,378,285),(124,440,379,284),(125,439,380,283),(126,438,381,282),(127,437,382,281),(128,436,383,280),(129,435,384,279),(130,434,361,278),(131,433,362,277),(132,456,363,276),(133,455,364,275),(134,454,365,274),(135,453,366,273),(136,452,367,272),(137,451,368,271),(138,450,369,270),(139,449,370,269),(140,448,371,268),(141,447,372,267),(142,446,373,266),(143,445,374,265),(144,444,375,288),(241,310,404,355),(242,309,405,354),(243,308,406,353),(244,307,407,352),(245,306,408,351),(246,305,385,350),(247,304,386,349),(248,303,387,348),(249,302,388,347),(250,301,389,346),(251,300,390,345),(252,299,391,344),(253,298,392,343),(254,297,393,342),(255,296,394,341),(256,295,395,340),(257,294,396,339),(258,293,397,338),(259,292,398,337),(260,291,399,360),(261,290,400,359),(262,289,401,358),(263,312,402,357),(264,311,403,356)]])

150 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 6A 6B 6C 8A 8B 8C 8D 10A ··· 10L 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 20I ··· 20X 24A ··· 24H 30A ··· 30L 40A ··· 40P 60A ··· 60P 120A ··· 120AF order 1 2 2 2 3 4 4 4 4 4 4 5 5 5 5 6 6 6 8 8 8 8 10 ··· 10 12 12 12 12 15 15 15 15 20 ··· 20 20 ··· 20 24 ··· 24 30 ··· 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 1 1 2 2 2 12 12 12 12 1 1 1 1 2 2 2 2 2 2 2 1 ··· 1 2 2 2 2 2 2 2 2 2 ··· 2 12 ··· 12 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

150 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - + + - - + + - image C1 C2 C2 C4 C5 C10 C10 C20 S3 Q8 D4 Dic3 D6 D8 Q16 Dic6 D12 C5×S3 C5×Q8 C5×D4 D24 Dic12 C5×Dic3 S3×C10 C5×D8 C5×Q16 C5×Dic6 C5×D12 C5×D24 C5×Dic12 kernel C5×C24⋊1C4 C5×C4⋊Dic3 C2×C120 C120 C24⋊1C4 C4⋊Dic3 C2×C24 C24 C2×C40 C60 C2×C30 C40 C2×C20 C30 C30 C20 C2×C10 C2×C8 C12 C2×C6 C10 C10 C8 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 1 4 4 8 4 16 1 1 1 2 1 2 2 2 2 4 4 4 4 4 8 4 8 8 8 8 16 16

Matrix representation of C5×C241C4 in GL4(𝔽241) generated by

 205 0 0 0 0 205 0 0 0 0 91 0 0 0 0 91
,
 240 1 0 0 240 0 0 0 0 0 9 136 0 0 105 114
,
 198 180 0 0 137 43 0 0 0 0 197 134 0 0 178 44
G:=sub<GL(4,GF(241))| [205,0,0,0,0,205,0,0,0,0,91,0,0,0,0,91],[240,240,0,0,1,0,0,0,0,0,9,105,0,0,136,114],[198,137,0,0,180,43,0,0,0,0,197,178,0,0,134,44] >;

C5×C241C4 in GAP, Magma, Sage, TeX

C_5\times C_{24}\rtimes_1C_4
% in TeX

G:=Group("C5xC24:1C4");
// GroupNames label

G:=SmallGroup(480,137);
// by ID

G=gap.SmallGroup(480,137);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,589,848,4204,102,15686]);
// Polycyclic

G:=Group<a,b,c|a^5=b^24=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

׿
×
𝔽