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## G = C2×C4×Dic15order 480 = 25·3·5

### Direct product of C2×C4 and Dic15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C2×C4×Dic15
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C2×Dic15 — C22×Dic15 — C2×C4×Dic15
 Lower central C15 — C2×C4×Dic15
 Upper central C1 — C22×C4

Generators and relations for C2×C4×Dic15
G = < a,b,c,d | a2=b4=c30=1, d2=c15, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 756 in 216 conjugacy classes, 135 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C42, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2×Dic3, C2×C12, C22×C6, C30, C30, C2×C42, C2×Dic5, C2×C20, C22×C10, C4×Dic3, C22×Dic3, C22×C12, Dic15, C60, C2×C30, C2×C30, C4×Dic5, C22×Dic5, C22×C20, C2×C4×Dic3, C2×Dic15, C2×C60, C22×C30, C2×C4×Dic5, C4×Dic15, C22×Dic15, C22×C60, C2×C4×Dic15
Quotients:

Smallest permutation representation of C2×C4×Dic15
Regular action on 480 points
Generators in S480
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 81)(16 82)(17 83)(18 84)(19 85)(20 86)(21 87)(22 88)(23 89)(24 90)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 160)(32 161)(33 162)(34 163)(35 164)(36 165)(37 166)(38 167)(39 168)(40 169)(41 170)(42 171)(43 172)(44 173)(45 174)(46 175)(47 176)(48 177)(49 178)(50 179)(51 180)(52 151)(53 152)(54 153)(55 154)(56 155)(57 156)(58 157)(59 158)(60 159)(91 480)(92 451)(93 452)(94 453)(95 454)(96 455)(97 456)(98 457)(99 458)(100 459)(101 460)(102 461)(103 462)(104 463)(105 464)(106 465)(107 466)(108 467)(109 468)(110 469)(111 470)(112 471)(113 472)(114 473)(115 474)(116 475)(117 476)(118 477)(119 478)(120 479)(121 287)(122 288)(123 289)(124 290)(125 291)(126 292)(127 293)(128 294)(129 295)(130 296)(131 297)(132 298)(133 299)(134 300)(135 271)(136 272)(137 273)(138 274)(139 275)(140 276)(141 277)(142 278)(143 279)(144 280)(145 281)(146 282)(147 283)(148 284)(149 285)(150 286)(181 235)(182 236)(183 237)(184 238)(185 239)(186 240)(187 211)(188 212)(189 213)(190 214)(191 215)(192 216)(193 217)(194 218)(195 219)(196 220)(197 221)(198 222)(199 223)(200 224)(201 225)(202 226)(203 227)(204 228)(205 229)(206 230)(207 231)(208 232)(209 233)(210 234)(241 444)(242 445)(243 446)(244 447)(245 448)(246 449)(247 450)(248 421)(249 422)(250 423)(251 424)(252 425)(253 426)(254 427)(255 428)(256 429)(257 430)(258 431)(259 432)(260 433)(261 434)(262 435)(263 436)(264 437)(265 438)(266 439)(267 440)(268 441)(269 442)(270 443)(301 395)(302 396)(303 397)(304 398)(305 399)(306 400)(307 401)(308 402)(309 403)(310 404)(311 405)(312 406)(313 407)(314 408)(315 409)(316 410)(317 411)(318 412)(319 413)(320 414)(321 415)(322 416)(323 417)(324 418)(325 419)(326 420)(327 391)(328 392)(329 393)(330 394)(331 378)(332 379)(333 380)(334 381)(335 382)(336 383)(337 384)(338 385)(339 386)(340 387)(341 388)(342 389)(343 390)(344 361)(345 362)(346 363)(347 364)(348 365)(349 366)(350 367)(351 368)(352 369)(353 370)(354 371)(355 372)(356 373)(357 374)(358 375)(359 376)(360 377)
(1 318 370 93)(2 319 371 94)(3 320 372 95)(4 321 373 96)(5 322 374 97)(6 323 375 98)(7 324 376 99)(8 325 377 100)(9 326 378 101)(10 327 379 102)(11 328 380 103)(12 329 381 104)(13 330 382 105)(14 301 383 106)(15 302 384 107)(16 303 385 108)(17 304 386 109)(18 305 387 110)(19 306 388 111)(20 307 389 112)(21 308 390 113)(22 309 361 114)(23 310 362 115)(24 311 363 116)(25 312 364 117)(26 313 365 118)(27 314 366 119)(28 315 367 120)(29 316 368 91)(30 317 369 92)(31 244 216 132)(32 245 217 133)(33 246 218 134)(34 247 219 135)(35 248 220 136)(36 249 221 137)(37 250 222 138)(38 251 223 139)(39 252 224 140)(40 253 225 141)(41 254 226 142)(42 255 227 143)(43 256 228 144)(44 257 229 145)(45 258 230 146)(46 259 231 147)(47 260 232 148)(48 261 233 149)(49 262 234 150)(50 263 235 121)(51 264 236 122)(52 265 237 123)(53 266 238 124)(54 267 239 125)(55 268 240 126)(56 269 211 127)(57 270 212 128)(58 241 213 129)(59 242 214 130)(60 243 215 131)(61 406 347 476)(62 407 348 477)(63 408 349 478)(64 409 350 479)(65 410 351 480)(66 411 352 451)(67 412 353 452)(68 413 354 453)(69 414 355 454)(70 415 356 455)(71 416 357 456)(72 417 358 457)(73 418 359 458)(74 419 360 459)(75 420 331 460)(76 391 332 461)(77 392 333 462)(78 393 334 463)(79 394 335 464)(80 395 336 465)(81 396 337 466)(82 397 338 467)(83 398 339 468)(84 399 340 469)(85 400 341 470)(86 401 342 471)(87 402 343 472)(88 403 344 473)(89 404 345 474)(90 405 346 475)(151 438 183 289)(152 439 184 290)(153 440 185 291)(154 441 186 292)(155 442 187 293)(156 443 188 294)(157 444 189 295)(158 445 190 296)(159 446 191 297)(160 447 192 298)(161 448 193 299)(162 449 194 300)(163 450 195 271)(164 421 196 272)(165 422 197 273)(166 423 198 274)(167 424 199 275)(168 425 200 276)(169 426 201 277)(170 427 202 278)(171 428 203 279)(172 429 204 280)(173 430 205 281)(174 431 206 282)(175 432 207 283)(176 433 208 284)(177 434 209 285)(178 435 210 286)(179 436 181 287)(180 437 182 288)
(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)
(1 292 16 277)(2 291 17 276)(3 290 18 275)(4 289 19 274)(5 288 20 273)(6 287 21 272)(7 286 22 271)(8 285 23 300)(9 284 24 299)(10 283 25 298)(11 282 26 297)(12 281 27 296)(13 280 28 295)(14 279 29 294)(15 278 30 293)(31 391 46 406)(32 420 47 405)(33 419 48 404)(34 418 49 403)(35 417 50 402)(36 416 51 401)(37 415 52 400)(38 414 53 399)(39 413 54 398)(40 412 55 397)(41 411 56 396)(42 410 57 395)(43 409 58 394)(44 408 59 393)(45 407 60 392)(61 132 76 147)(62 131 77 146)(63 130 78 145)(64 129 79 144)(65 128 80 143)(66 127 81 142)(67 126 82 141)(68 125 83 140)(69 124 84 139)(70 123 85 138)(71 122 86 137)(72 121 87 136)(73 150 88 135)(74 149 89 134)(75 148 90 133)(91 188 106 203)(92 187 107 202)(93 186 108 201)(94 185 109 200)(95 184 110 199)(96 183 111 198)(97 182 112 197)(98 181 113 196)(99 210 114 195)(100 209 115 194)(101 208 116 193)(102 207 117 192)(103 206 118 191)(104 205 119 190)(105 204 120 189)(151 306 166 321)(152 305 167 320)(153 304 168 319)(154 303 169 318)(155 302 170 317)(156 301 171 316)(157 330 172 315)(158 329 173 314)(159 328 174 313)(160 327 175 312)(161 326 176 311)(162 325 177 310)(163 324 178 309)(164 323 179 308)(165 322 180 307)(211 466 226 451)(212 465 227 480)(213 464 228 479)(214 463 229 478)(215 462 230 477)(216 461 231 476)(217 460 232 475)(218 459 233 474)(219 458 234 473)(220 457 235 472)(221 456 236 471)(222 455 237 470)(223 454 238 469)(224 453 239 468)(225 452 240 467)(241 335 256 350)(242 334 257 349)(243 333 258 348)(244 332 259 347)(245 331 260 346)(246 360 261 345)(247 359 262 344)(248 358 263 343)(249 357 264 342)(250 356 265 341)(251 355 266 340)(252 354 267 339)(253 353 268 338)(254 352 269 337)(255 351 270 336)(361 450 376 435)(362 449 377 434)(363 448 378 433)(364 447 379 432)(365 446 380 431)(366 445 381 430)(367 444 382 429)(368 443 383 428)(369 442 384 427)(370 441 385 426)(371 440 386 425)(372 439 387 424)(373 438 388 423)(374 437 389 422)(375 436 390 421)

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

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

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

144 conjugacy classes

 class 1 2A ··· 2G 3 4A ··· 4H 4I ··· 4X 5A 5B 6A ··· 6G 10A ··· 10N 12A ··· 12H 15A 15B 15C 15D 20A ··· 20P 30A ··· 30AB 60A ··· 60AF order 1 2 ··· 2 3 4 ··· 4 4 ··· 4 5 5 6 ··· 6 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 ··· 1 2 1 ··· 1 15 ··· 15 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

144 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + + - + + + - + + image C1 C2 C2 C2 C4 C4 S3 D5 Dic3 D6 D6 Dic5 D10 D10 C4×S3 D15 C4×D5 Dic15 D30 D30 C4×D15 kernel C2×C4×Dic15 C4×Dic15 C22×Dic15 C22×C60 C2×Dic15 C2×C60 C22×C20 C22×C12 C2×C20 C2×C20 C22×C10 C2×C12 C2×C12 C22×C6 C2×C10 C22×C4 C2×C6 C2×C4 C2×C4 C23 C22 # reps 1 4 2 1 16 8 1 2 4 2 1 8 4 2 8 4 16 16 8 4 32

Matrix representation of C2×C4×Dic15 in GL5(𝔽61)

 60 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 60 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 11
,
 1 0 0 0 0 0 0 60 0 0 0 1 44 0 0 0 0 0 28 37 0 0 0 24 47
,
 1 0 0 0 0 0 36 30 0 0 0 32 25 0 0 0 0 0 20 29 0 0 0 3 41

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,11],[1,0,0,0,0,0,0,1,0,0,0,60,44,0,0,0,0,0,28,24,0,0,0,37,47],[1,0,0,0,0,0,36,32,0,0,0,30,25,0,0,0,0,0,20,3,0,0,0,29,41] >;

C2×C4×Dic15 in GAP, Magma, Sage, TeX

C_2\times C_4\times {\rm Dic}_{15}
% in TeX

G:=Group("C2xC4xDic15");
// GroupNames label

G:=SmallGroup(480,887);
// by ID

G=gap.SmallGroup(480,887);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,100,2693,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^30=1,d^2=c^15,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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