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G = C5×Dic24order 480 = 25·3·5

Direct product of C5 and Dic24

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

Aliases: C5×Dic24, C156Q32, C80.2S3, C48.1C10, C240.3C2, C40.77D6, C30.34D8, C20.41D12, C60.179D4, C10.15D24, C120.95C22, Dic12.1C10, C16.(C5×S3), C31(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

C1C24 — C5×Dic24
C1C3C6C12C24C120C5×Dic12 — C5×Dic24
C3C6C12C24 — C5×Dic24
C1C10C20C40C80

Generators and relations for C5×Dic24
 G = < a,b,c | a5=b48=1, c2=b24, ab=ba, ac=ca, cbc-1=b-1 >

12C4
12C4
6Q8
6Q8
4Dic3
4Dic3
12C20
12C20
3Q16
3Q16
2Dic6
2Dic6
6C5×Q8
6C5×Q8
4C5×Dic3
4C5×Dic3
3Q32
3C5×Q16
3C5×Q16
2C5×Dic6
2C5×Dic6
3C5×Q32

Smallest permutation representation of C5×Dic24
Regular action on 480 points
Generators in S480
(1 384 131 240 420)(2 337 132 193 421)(3 338 133 194 422)(4 339 134 195 423)(5 340 135 196 424)(6 341 136 197 425)(7 342 137 198 426)(8 343 138 199 427)(9 344 139 200 428)(10 345 140 201 429)(11 346 141 202 430)(12 347 142 203 431)(13 348 143 204 432)(14 349 144 205 385)(15 350 97 206 386)(16 351 98 207 387)(17 352 99 208 388)(18 353 100 209 389)(19 354 101 210 390)(20 355 102 211 391)(21 356 103 212 392)(22 357 104 213 393)(23 358 105 214 394)(24 359 106 215 395)(25 360 107 216 396)(26 361 108 217 397)(27 362 109 218 398)(28 363 110 219 399)(29 364 111 220 400)(30 365 112 221 401)(31 366 113 222 402)(32 367 114 223 403)(33 368 115 224 404)(34 369 116 225 405)(35 370 117 226 406)(36 371 118 227 407)(37 372 119 228 408)(38 373 120 229 409)(39 374 121 230 410)(40 375 122 231 411)(41 376 123 232 412)(42 377 124 233 413)(43 378 125 234 414)(44 379 126 235 415)(45 380 127 236 416)(46 381 128 237 417)(47 382 129 238 418)(48 383 130 239 419)(49 253 153 300 469)(50 254 154 301 470)(51 255 155 302 471)(52 256 156 303 472)(53 257 157 304 473)(54 258 158 305 474)(55 259 159 306 475)(56 260 160 307 476)(57 261 161 308 477)(58 262 162 309 478)(59 263 163 310 479)(60 264 164 311 480)(61 265 165 312 433)(62 266 166 313 434)(63 267 167 314 435)(64 268 168 315 436)(65 269 169 316 437)(66 270 170 317 438)(67 271 171 318 439)(68 272 172 319 440)(69 273 173 320 441)(70 274 174 321 442)(71 275 175 322 443)(72 276 176 323 444)(73 277 177 324 445)(74 278 178 325 446)(75 279 179 326 447)(76 280 180 327 448)(77 281 181 328 449)(78 282 182 329 450)(79 283 183 330 451)(80 284 184 331 452)(81 285 185 332 453)(82 286 186 333 454)(83 287 187 334 455)(84 288 188 335 456)(85 241 189 336 457)(86 242 190 289 458)(87 243 191 290 459)(88 244 192 291 460)(89 245 145 292 461)(90 246 146 293 462)(91 247 147 294 463)(92 248 148 295 464)(93 249 149 296 465)(94 250 150 297 466)(95 251 151 298 467)(96 252 152 299 468)
(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)
(1 320 25 296)(2 319 26 295)(3 318 27 294)(4 317 28 293)(5 316 29 292)(6 315 30 291)(7 314 31 290)(8 313 32 289)(9 312 33 336)(10 311 34 335)(11 310 35 334)(12 309 36 333)(13 308 37 332)(14 307 38 331)(15 306 39 330)(16 305 40 329)(17 304 41 328)(18 303 42 327)(19 302 43 326)(20 301 44 325)(21 300 45 324)(22 299 46 323)(23 298 47 322)(24 297 48 321)(49 127 73 103)(50 126 74 102)(51 125 75 101)(52 124 76 100)(53 123 77 99)(54 122 78 98)(55 121 79 97)(56 120 80 144)(57 119 81 143)(58 118 82 142)(59 117 83 141)(60 116 84 140)(61 115 85 139)(62 114 86 138)(63 113 87 137)(64 112 88 136)(65 111 89 135)(66 110 90 134)(67 109 91 133)(68 108 92 132)(69 107 93 131)(70 106 94 130)(71 105 95 129)(72 104 96 128)(145 424 169 400)(146 423 170 399)(147 422 171 398)(148 421 172 397)(149 420 173 396)(150 419 174 395)(151 418 175 394)(152 417 176 393)(153 416 177 392)(154 415 178 391)(155 414 179 390)(156 413 180 389)(157 412 181 388)(158 411 182 387)(159 410 183 386)(160 409 184 385)(161 408 185 432)(162 407 186 431)(163 406 187 430)(164 405 188 429)(165 404 189 428)(166 403 190 427)(167 402 191 426)(168 401 192 425)(193 272 217 248)(194 271 218 247)(195 270 219 246)(196 269 220 245)(197 268 221 244)(198 267 222 243)(199 266 223 242)(200 265 224 241)(201 264 225 288)(202 263 226 287)(203 262 227 286)(204 261 228 285)(205 260 229 284)(206 259 230 283)(207 258 231 282)(208 257 232 281)(209 256 233 280)(210 255 234 279)(211 254 235 278)(212 253 236 277)(213 252 237 276)(214 251 238 275)(215 250 239 274)(216 249 240 273)(337 440 361 464)(338 439 362 463)(339 438 363 462)(340 437 364 461)(341 436 365 460)(342 435 366 459)(343 434 367 458)(344 433 368 457)(345 480 369 456)(346 479 370 455)(347 478 371 454)(348 477 372 453)(349 476 373 452)(350 475 374 451)(351 474 375 450)(352 473 376 449)(353 472 377 448)(354 471 378 447)(355 470 379 446)(356 469 380 445)(357 468 381 444)(358 467 382 443)(359 466 383 442)(360 465 384 441)

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

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

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

135 conjugacy classes

class 1  2  3 4A4B4C5A5B5C5D 6 8A8B10A10B10C10D12A12B15A15B15C15D16A16B16C16D20A20B20C20D20E···20L24A24B24C24D30A30B30C30D40A···40H48A···48H60A···60H80A···80P120A···120P240A···240AF
order123444555568810101010121215151515161616162020202020···20242424243030303040···4048···4860···6080···80120···120240···240
size11222424111122211112222222222222224···24222222222···22···22···22···22···22···2

135 irreducible representations

dim1111112222222222222222
type++++++++-+-
imageC1C2C2C5C10C10S3D4D6D8D12C5×S3Q32C5×D4D24S3×C10C5×D8Dic24C5×D12C5×Q32C5×D24C5×Dic24
kernelC5×Dic24C240C5×Dic12Dic24C48Dic12C80C60C40C30C20C16C15C12C10C8C6C5C4C3C2C1
# reps1124481112244444888161632

Matrix representation of C5×Dic24 in GL4(𝔽241) generated by

205000
020500
0010
0001
,
966600
13716700
0044178
0063107
,
2052900
803600
00116207
0091125
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

Export

Subgroup lattice of C5×Dic24 in TeX

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