Copied to
clipboard

G = C5×C241C4order 480 = 25·3·5

Direct product of C5 and C241C4

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

Aliases: C5×C241C4, C241C20, C12014C4, C409Dic3, C30.35D8, C60.31Q8, C10.16D24, C30.15Q16, C20.25Dic6, C10.7Dic12, C6.4(C5×D8), C81(C5×Dic3), C2.1(C5×D24), C6.2(C5×Q16), C12.5(C5×Q8), (C2×C40).13S3, (C2×C24).5C10, C1512(C2.D8), C30.58(C4⋊C4), C4.5(C5×Dic6), (C2×C120).20C2, C60.245(C2×C4), C12.35(C2×C20), (C2×C10).50D12, (C2×C30).120D4, (C2×C20).422D6, C4⋊Dic3.3C10, C2.2(C5×Dic12), C4.7(C10×Dic3), C22.9(C5×D12), C20.67(C2×Dic3), (C2×C60).523C22, C10.20(C4⋊Dic3), C32(C5×C2.D8), C6.6(C5×C4⋊C4), (C2×C8).3(C5×S3), (C2×C6).14(C5×D4), C2.4(C5×C4⋊Dic3), (C2×C4).72(S3×C10), (C2×C12).87(C2×C10), (C5×C4⋊Dic3).15C2, SmallGroup(480,137)

Series: Derived Chief Lower central Upper central

C1C12 — C5×C241C4
C1C3C6C12C2×C12C2×C60C5×C4⋊Dic3 — C5×C241C4
C3C6C12 — C5×C241C4
C1C2×C10C2×C20C2×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 [×3], C3, C4 [×2], C4 [×2], C22, C5, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C10 [×3], Dic3 [×2], C12 [×2], C2×C6, C15, C4⋊C4 [×2], C2×C8, C20 [×2], C20 [×2], C2×C10, C24 [×2], C2×Dic3 [×2], C2×C12, C30 [×3], C2.D8, C40 [×2], C2×C20, C2×C20 [×2], C4⋊Dic3 [×2], C2×C24, C5×Dic3 [×2], C60 [×2], C2×C30, C5×C4⋊C4 [×2], C2×C40, C241C4, C120 [×2], C10×Dic3 [×2], C2×C60, C5×C2.D8, C5×C4⋊Dic3 [×2], C2×C120, C5×C241C4
Quotients: C1, C2 [×3], C4 [×2], C22, C5, S3, C2×C4, D4, Q8, C10 [×3], Dic3 [×2], D6, C4⋊C4, D8, Q16, C20 [×2], C2×C10, Dic6, D12, C2×Dic3, C5×S3, C2.D8, C2×C20, C5×D4, C5×Q8, D24, Dic12, C4⋊Dic3, C5×Dic3 [×2], 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
(1 353 292 211 79)(2 354 293 212 80)(3 355 294 213 81)(4 356 295 214 82)(5 357 296 215 83)(6 358 297 216 84)(7 359 298 193 85)(8 360 299 194 86)(9 337 300 195 87)(10 338 301 196 88)(11 339 302 197 89)(12 340 303 198 90)(13 341 304 199 91)(14 342 305 200 92)(15 343 306 201 93)(16 344 307 202 94)(17 345 308 203 95)(18 346 309 204 96)(19 347 310 205 73)(20 348 311 206 74)(21 349 312 207 75)(22 350 289 208 76)(23 351 290 209 77)(24 352 291 210 78)(25 468 401 423 276)(26 469 402 424 277)(27 470 403 425 278)(28 471 404 426 279)(29 472 405 427 280)(30 473 406 428 281)(31 474 407 429 282)(32 475 408 430 283)(33 476 385 431 284)(34 477 386 432 285)(35 478 387 409 286)(36 479 388 410 287)(37 480 389 411 288)(38 457 390 412 265)(39 458 391 413 266)(40 459 392 414 267)(41 460 393 415 268)(42 461 394 416 269)(43 462 395 417 270)(44 463 396 418 271)(45 464 397 419 272)(46 465 398 420 273)(47 466 399 421 274)(48 467 400 422 275)(49 315 240 440 142)(50 316 217 441 143)(51 317 218 442 144)(52 318 219 443 121)(53 319 220 444 122)(54 320 221 445 123)(55 321 222 446 124)(56 322 223 447 125)(57 323 224 448 126)(58 324 225 449 127)(59 325 226 450 128)(60 326 227 451 129)(61 327 228 452 130)(62 328 229 453 131)(63 329 230 454 132)(64 330 231 455 133)(65 331 232 456 134)(66 332 233 433 135)(67 333 234 434 136)(68 334 235 435 137)(69 335 236 436 138)(70 336 237 437 139)(71 313 238 438 140)(72 314 239 439 141)(97 174 253 384 148)(98 175 254 361 149)(99 176 255 362 150)(100 177 256 363 151)(101 178 257 364 152)(102 179 258 365 153)(103 180 259 366 154)(104 181 260 367 155)(105 182 261 368 156)(106 183 262 369 157)(107 184 263 370 158)(108 185 264 371 159)(109 186 241 372 160)(110 187 242 373 161)(111 188 243 374 162)(112 189 244 375 163)(113 190 245 376 164)(114 191 246 377 165)(115 192 247 378 166)(116 169 248 379 167)(117 170 249 380 168)(118 171 250 381 145)(119 172 251 382 146)(120 173 252 383 147)
(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 64 187 431)(2 63 188 430)(3 62 189 429)(4 61 190 428)(5 60 191 427)(6 59 192 426)(7 58 169 425)(8 57 170 424)(9 56 171 423)(10 55 172 422)(11 54 173 421)(12 53 174 420)(13 52 175 419)(14 51 176 418)(15 50 177 417)(16 49 178 416)(17 72 179 415)(18 71 180 414)(19 70 181 413)(20 69 182 412)(21 68 183 411)(22 67 184 410)(23 66 185 409)(24 65 186 432)(25 300 223 381)(26 299 224 380)(27 298 225 379)(28 297 226 378)(29 296 227 377)(30 295 228 376)(31 294 229 375)(32 293 230 374)(33 292 231 373)(34 291 232 372)(35 290 233 371)(36 289 234 370)(37 312 235 369)(38 311 236 368)(39 310 237 367)(40 309 238 366)(41 308 239 365)(42 307 240 364)(43 306 217 363)(44 305 218 362)(45 304 219 361)(46 303 220 384)(47 302 221 383)(48 301 222 382)(73 139 104 391)(74 138 105 390)(75 137 106 389)(76 136 107 388)(77 135 108 387)(78 134 109 386)(79 133 110 385)(80 132 111 408)(81 131 112 407)(82 130 113 406)(83 129 114 405)(84 128 115 404)(85 127 116 403)(86 126 117 402)(87 125 118 401)(88 124 119 400)(89 123 120 399)(90 122 97 398)(91 121 98 397)(92 144 99 396)(93 143 100 395)(94 142 101 394)(95 141 102 393)(96 140 103 392)(145 468 195 447)(146 467 196 446)(147 466 197 445)(148 465 198 444)(149 464 199 443)(150 463 200 442)(151 462 201 441)(152 461 202 440)(153 460 203 439)(154 459 204 438)(155 458 205 437)(156 457 206 436)(157 480 207 435)(158 479 208 434)(159 478 209 433)(160 477 210 456)(161 476 211 455)(162 475 212 454)(163 474 213 453)(164 473 214 452)(165 472 215 451)(166 471 216 450)(167 470 193 449)(168 469 194 448)(241 285 352 331)(242 284 353 330)(243 283 354 329)(244 282 355 328)(245 281 356 327)(246 280 357 326)(247 279 358 325)(248 278 359 324)(249 277 360 323)(250 276 337 322)(251 275 338 321)(252 274 339 320)(253 273 340 319)(254 272 341 318)(255 271 342 317)(256 270 343 316)(257 269 344 315)(258 268 345 314)(259 267 346 313)(260 266 347 336)(261 265 348 335)(262 288 349 334)(263 287 350 333)(264 286 351 332)

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

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

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

150 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B5C5D6A6B6C8A8B8C8D10A···10L12A12B12C12D15A15B15C15D20A···20H20I···20X24A···24H30A···30L40A···40P60A···60P120A···120AF
order122234444445555666888810···10121212121515151520···2020···2024···2430···3040···4060···60120···120
size111122212121212111122222221···1222222222···212···122···22···22···22···22···2

150 irreducible representations

dim111111112222222222222222222222
type++++-+-++--++-
imageC1C2C2C4C5C10C10C20S3Q8D4Dic3D6D8Q16Dic6D12C5×S3C5×Q8C5×D4D24Dic12C5×Dic3S3×C10C5×D8C5×Q16C5×Dic6C5×D12C5×D24C5×Dic12
kernelC5×C241C4C5×C4⋊Dic3C2×C120C120C241C4C4⋊Dic3C2×C24C24C2×C40C60C2×C30C40C2×C20C30C30C20C2×C10C2×C8C12C2×C6C10C10C8C2×C4C6C6C4C22C2C2
# reps121448416111212222444448488881616

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

205000
020500
00910
00091
,
240100
240000
009136
00105114
,
19818000
1374300
00197134
0017844
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