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## G = C5×C24⋊C4order 480 = 25·3·5

### Direct product of C5 and C24⋊C4

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5×C24⋊C4
G = < a,b,c | a5=b24=c4=1, ab=ba, ac=ca, cbc-1=b5 >

Subgroups: 132 in 80 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, C2×C4, C10, C10, Dic3, C12, C2×C6, C15, C42, C2×C8, C2×C8, C20, C20, C2×C10, C3⋊C8, C24, C2×Dic3, C2×C12, C30, C30, C8⋊C4, C40, C40, C2×C20, C2×C20, C2×C3⋊C8, C4×Dic3, C2×C24, C5×Dic3, C60, C2×C30, C4×C20, C2×C40, C2×C40, C24⋊C4, C5×C3⋊C8, C120, C10×Dic3, C2×C60, C5×C8⋊C4, C10×C3⋊C8, Dic3×C20, C2×C120, C5×C24⋊C4
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C10, Dic3, D6, C42, M4(2), C20, C2×C10, C4×S3, C2×Dic3, C5×S3, C8⋊C4, C2×C20, C8⋊S3, C4×Dic3, C5×Dic3, S3×C10, C4×C20, C5×M4(2), C24⋊C4, S3×C20, C10×Dic3, C5×C8⋊C4, C5×C8⋊S3, Dic3×C20, C5×C24⋊C4

Smallest permutation representation of C5×C24⋊C4
Regular action on 480 points
Generators in S480
(1 277 240 196 153)(2 278 217 197 154)(3 279 218 198 155)(4 280 219 199 156)(5 281 220 200 157)(6 282 221 201 158)(7 283 222 202 159)(8 284 223 203 160)(9 285 224 204 161)(10 286 225 205 162)(11 287 226 206 163)(12 288 227 207 164)(13 265 228 208 165)(14 266 229 209 166)(15 267 230 210 167)(16 268 231 211 168)(17 269 232 212 145)(18 270 233 213 146)(19 271 234 214 147)(20 272 235 215 148)(21 273 236 216 149)(22 274 237 193 150)(23 275 238 194 151)(24 276 239 195 152)(25 394 260 377 106)(26 395 261 378 107)(27 396 262 379 108)(28 397 263 380 109)(29 398 264 381 110)(30 399 241 382 111)(31 400 242 383 112)(32 401 243 384 113)(33 402 244 361 114)(34 403 245 362 115)(35 404 246 363 116)(36 405 247 364 117)(37 406 248 365 118)(38 407 249 366 119)(39 408 250 367 120)(40 385 251 368 97)(41 386 252 369 98)(42 387 253 370 99)(43 388 254 371 100)(44 389 255 372 101)(45 390 256 373 102)(46 391 257 374 103)(47 392 258 375 104)(48 393 259 376 105)(49 96 180 300 411)(50 73 181 301 412)(51 74 182 302 413)(52 75 183 303 414)(53 76 184 304 415)(54 77 185 305 416)(55 78 186 306 417)(56 79 187 307 418)(57 80 188 308 419)(58 81 189 309 420)(59 82 190 310 421)(60 83 191 311 422)(61 84 192 312 423)(62 85 169 289 424)(63 86 170 290 425)(64 87 171 291 426)(65 88 172 292 427)(66 89 173 293 428)(67 90 174 294 429)(68 91 175 295 430)(69 92 176 296 431)(70 93 177 297 432)(71 94 178 298 409)(72 95 179 299 410)(121 435 323 461 338)(122 436 324 462 339)(123 437 325 463 340)(124 438 326 464 341)(125 439 327 465 342)(126 440 328 466 343)(127 441 329 467 344)(128 442 330 468 345)(129 443 331 469 346)(130 444 332 470 347)(131 445 333 471 348)(132 446 334 472 349)(133 447 335 473 350)(134 448 336 474 351)(135 449 313 475 352)(136 450 314 476 353)(137 451 315 477 354)(138 452 316 478 355)(139 453 317 479 356)(140 454 318 480 357)(141 455 319 457 358)(142 456 320 458 359)(143 433 321 459 360)(144 434 322 460 337)
(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)
(1 172 368 455)(2 177 369 436)(3 182 370 441)(4 187 371 446)(5 192 372 451)(6 173 373 456)(7 178 374 437)(8 183 375 442)(9 188 376 447)(10 169 377 452)(11 174 378 433)(12 179 379 438)(13 184 380 443)(14 189 381 448)(15 170 382 453)(16 175 383 434)(17 180 384 439)(18 185 361 444)(19 190 362 449)(20 171 363 454)(21 176 364 435)(22 181 365 440)(23 186 366 445)(24 191 367 450)(25 478 225 424)(26 459 226 429)(27 464 227 410)(28 469 228 415)(29 474 229 420)(30 479 230 425)(31 460 231 430)(32 465 232 411)(33 470 233 416)(34 475 234 421)(35 480 235 426)(36 461 236 431)(37 466 237 412)(38 471 238 417)(39 476 239 422)(40 457 240 427)(41 462 217 432)(42 467 218 413)(43 472 219 418)(44 477 220 423)(45 458 221 428)(46 463 222 409)(47 468 223 414)(48 473 224 419)(49 401 342 212)(50 406 343 193)(51 387 344 198)(52 392 345 203)(53 397 346 208)(54 402 347 213)(55 407 348 194)(56 388 349 199)(57 393 350 204)(58 398 351 209)(59 403 352 214)(60 408 353 195)(61 389 354 200)(62 394 355 205)(63 399 356 210)(64 404 357 215)(65 385 358 196)(66 390 359 201)(67 395 360 206)(68 400 337 211)(69 405 338 216)(70 386 339 197)(71 391 340 202)(72 396 341 207)(73 248 126 150)(74 253 127 155)(75 258 128 160)(76 263 129 165)(77 244 130 146)(78 249 131 151)(79 254 132 156)(80 259 133 161)(81 264 134 166)(82 245 135 147)(83 250 136 152)(84 255 137 157)(85 260 138 162)(86 241 139 167)(87 246 140 148)(88 251 141 153)(89 256 142 158)(90 261 143 163)(91 242 144 168)(92 247 121 149)(93 252 122 154)(94 257 123 159)(95 262 124 164)(96 243 125 145)(97 319 277 292)(98 324 278 297)(99 329 279 302)(100 334 280 307)(101 315 281 312)(102 320 282 293)(103 325 283 298)(104 330 284 303)(105 335 285 308)(106 316 286 289)(107 321 287 294)(108 326 288 299)(109 331 265 304)(110 336 266 309)(111 317 267 290)(112 322 268 295)(113 327 269 300)(114 332 270 305)(115 313 271 310)(116 318 272 291)(117 323 273 296)(118 328 274 301)(119 333 275 306)(120 314 276 311)

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

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

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

180 conjugacy classes

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

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C2 C4 C4 C4 C5 C10 C10 C10 C20 C20 C20 S3 Dic3 D6 M4(2) C4×S3 C4×S3 C5×S3 C8⋊S3 C5×Dic3 S3×C10 C5×M4(2) S3×C20 S3×C20 C5×C8⋊S3 kernel C5×C24⋊C4 C10×C3⋊C8 Dic3×C20 C2×C120 C5×C3⋊C8 C120 C10×Dic3 C24⋊C4 C2×C3⋊C8 C4×Dic3 C2×C24 C3⋊C8 C24 C2×Dic3 C2×C40 C40 C2×C20 C30 C20 C2×C10 C2×C8 C10 C8 C2×C4 C6 C4 C22 C2 # reps 1 1 1 1 4 4 4 4 4 4 4 16 16 16 1 2 1 4 2 2 4 8 8 4 16 8 8 32

Matrix representation of C5×C24⋊C4 in GL4(𝔽241) generated by

 87 0 0 0 0 87 0 0 0 0 87 0 0 0 0 87
,
 0 1 0 0 240 240 0 0 0 0 85 163 0 0 78 163
,
 113 199 0 0 86 128 0 0 0 0 65 165 0 0 100 176
G:=sub<GL(4,GF(241))| [87,0,0,0,0,87,0,0,0,0,87,0,0,0,0,87],[0,240,0,0,1,240,0,0,0,0,85,78,0,0,163,163],[113,86,0,0,199,128,0,0,0,0,65,100,0,0,165,176] >;

C5×C24⋊C4 in GAP, Magma, Sage, TeX

C_5\times C_{24}\rtimes C_4
% in TeX

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

G:=SmallGroup(480,134);
// by ID

G=gap.SmallGroup(480,134);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,1149,288,102,15686]);
// 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^5>;
// generators/relations

׿
×
𝔽