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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 [×6], C3, C4 [×4], C4 [×8], C22, C22 [×6], C5, C6, C6 [×6], C2×C4 [×6], C2×C4 [×12], C23, C10, C10 [×6], Dic3 [×8], C12 [×4], C2×C6, C2×C6 [×6], C15, C42 [×4], C22×C4, C22×C4 [×2], Dic5 [×8], C20 [×4], C2×C10, C2×C10 [×6], C2×Dic3 [×12], C2×C12 [×6], C22×C6, C30, C30 [×6], C2×C42, C2×Dic5 [×12], C2×C20 [×6], C22×C10, C4×Dic3 [×4], C22×Dic3 [×2], C22×C12, Dic15 [×8], C60 [×4], C2×C30, C2×C30 [×6], C4×Dic5 [×4], C22×Dic5 [×2], C22×C20, C2×C4×Dic3, C2×Dic15 [×12], C2×C60 [×6], C22×C30, C2×C4×Dic5, C4×Dic15 [×4], C22×Dic15 [×2], C22×C60, C2×C4×Dic15
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], S3, C2×C4 [×18], C23, D5, Dic3 [×4], D6 [×3], C42 [×4], C22×C4 [×3], Dic5 [×4], D10 [×3], C4×S3 [×4], C2×Dic3 [×6], C22×S3, D15, C2×C42, C4×D5 [×4], C2×Dic5 [×6], C22×D5, C4×Dic3 [×4], S3×C2×C4 [×2], C22×Dic3, Dic15 [×4], D30 [×3], C4×Dic5 [×4], C2×C4×D5 [×2], C22×Dic5, C2×C4×Dic3, C4×D15 [×4], C2×Dic15 [×6], C22×D15, C2×C4×Dic5, C4×Dic15 [×4], C2×C4×D15 [×2], C22×Dic15, C2×C4×Dic15

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

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

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

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

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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