direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C15×C42.C2, C60.39Q8, C42.4C30, C4⋊C4.4C30, C2.4(Q8×C30), C4.3(Q8×C15), (C4×C20).10C6, (C4×C60).22C2, C20.11(C3×Q8), C10.21(C6×Q8), C12.11(C5×Q8), C6.21(Q8×C10), (C4×C12).10C10, C30.119(C2×Q8), C30.281(C4○D4), (C2×C30).460C23, (C2×C60).470C22, C22.15(C22×C30), (C5×C4⋊C4).11C6, C2.8(C15×C4○D4), C6.45(C5×C4○D4), (C3×C4⋊C4).11C10, (C15×C4⋊C4).25C2, (C2×C20).83(C2×C6), (C2×C4).14(C2×C30), C10.45(C3×C4○D4), (C2×C12).83(C2×C10), (C2×C10).80(C22×C6), (C2×C6).80(C22×C10), SmallGroup(480,930)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×C42.C2
G = < a,b,c,d | a15=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >
Subgroups: 136 in 112 conjugacy classes, 88 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, C2×C4, C10, C10, C12, C12, C2×C6, C15, C42, C4⋊C4, C20, C20, C2×C10, C2×C12, C2×C12, C30, C30, C42.C2, C2×C20, C2×C20, C4×C12, C3×C4⋊C4, C60, C60, C2×C30, C4×C20, C5×C4⋊C4, C3×C42.C2, C2×C60, C2×C60, C5×C42.C2, C4×C60, C15×C4⋊C4, C15×C42.C2
Quotients: C1, C2, C3, C22, C5, C6, Q8, C23, C10, C2×C6, C15, C2×Q8, C4○D4, C2×C10, C3×Q8, C22×C6, C30, C42.C2, C5×Q8, C22×C10, C6×Q8, C3×C4○D4, C2×C30, Q8×C10, C5×C4○D4, C3×C42.C2, Q8×C15, C22×C30, C5×C42.C2, Q8×C30, C15×C4○D4, C15×C42.C2
►Regular action on 480 points
Generators in S480
(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)
(1 100 307 259)(2 101 308 260)(3 102 309 261)(4 103 310 262)(5 104 311 263)(6 105 312 264)(7 91 313 265)(8 92 314 266)(9 93 315 267)(10 94 301 268)(11 95 302 269)(12 96 303 270)(13 97 304 256)(14 98 305 257)(15 99 306 258)(16 113 221 444)(17 114 222 445)(18 115 223 446)(19 116 224 447)(20 117 225 448)(21 118 211 449)(22 119 212 450)(23 120 213 436)(24 106 214 437)(25 107 215 438)(26 108 216 439)(27 109 217 440)(28 110 218 441)(29 111 219 442)(30 112 220 443)(31 279 350 171)(32 280 351 172)(33 281 352 173)(34 282 353 174)(35 283 354 175)(36 284 355 176)(37 285 356 177)(38 271 357 178)(39 272 358 179)(40 273 359 180)(41 274 360 166)(42 275 346 167)(43 276 347 168)(44 277 348 169)(45 278 349 170)(46 474 298 154)(47 475 299 155)(48 476 300 156)(49 477 286 157)(50 478 287 158)(51 479 288 159)(52 480 289 160)(53 466 290 161)(54 467 291 162)(55 468 292 163)(56 469 293 164)(57 470 294 165)(58 471 295 151)(59 472 296 152)(60 473 297 153)(61 199 337 406)(62 200 338 407)(63 201 339 408)(64 202 340 409)(65 203 341 410)(66 204 342 411)(67 205 343 412)(68 206 344 413)(69 207 345 414)(70 208 331 415)(71 209 332 416)(72 210 333 417)(73 196 334 418)(74 197 335 419)(75 198 336 420)(76 459 125 394)(77 460 126 395)(78 461 127 396)(79 462 128 397)(80 463 129 398)(81 464 130 399)(82 465 131 400)(83 451 132 401)(84 452 133 402)(85 453 134 403)(86 454 135 404)(87 455 121 405)(88 456 122 391)(89 457 123 392)(90 458 124 393)(136 184 229 422)(137 185 230 423)(138 186 231 424)(139 187 232 425)(140 188 233 426)(141 189 234 427)(142 190 235 428)(143 191 236 429)(144 192 237 430)(145 193 238 431)(146 194 239 432)(147 195 240 433)(148 181 226 434)(149 182 227 435)(150 183 228 421)(241 368 377 326)(242 369 378 327)(243 370 379 328)(244 371 380 329)(245 372 381 330)(246 373 382 316)(247 374 383 317)(248 375 384 318)(249 361 385 319)(250 362 386 320)(251 363 387 321)(252 364 388 322)(253 365 389 323)(254 366 390 324)(255 367 376 325)
(1 80 139 338)(2 81 140 339)(3 82 141 340)(4 83 142 341)(5 84 143 342)(6 85 144 343)(7 86 145 344)(8 87 146 345)(9 88 147 331)(10 89 148 332)(11 90 149 333)(12 76 150 334)(13 77 136 335)(14 78 137 336)(15 79 138 337)(16 320 347 47)(17 321 348 48)(18 322 349 49)(19 323 350 50)(20 324 351 51)(21 325 352 52)(22 326 353 53)(23 327 354 54)(24 328 355 55)(25 329 356 56)(26 330 357 57)(27 316 358 58)(28 317 359 59)(29 318 360 60)(30 319 346 46)(31 287 224 365)(32 288 225 366)(33 289 211 367)(34 290 212 368)(35 291 213 369)(36 292 214 370)(37 293 215 371)(38 294 216 372)(39 295 217 373)(40 296 218 374)(41 297 219 375)(42 298 220 361)(43 299 221 362)(44 300 222 363)(45 286 223 364)(61 306 128 231)(62 307 129 232)(63 308 130 233)(64 309 131 234)(65 310 132 235)(66 311 133 236)(67 312 134 237)(68 313 135 238)(69 314 121 239)(70 315 122 240)(71 301 123 226)(72 302 124 227)(73 303 125 228)(74 304 126 229)(75 305 127 230)(91 454 193 413)(92 455 194 414)(93 456 195 415)(94 457 181 416)(95 458 182 417)(96 459 183 418)(97 460 184 419)(98 461 185 420)(99 462 186 406)(100 463 187 407)(101 464 188 408)(102 465 189 409)(103 451 190 410)(104 452 191 411)(105 453 192 412)(106 243 176 468)(107 244 177 469)(108 245 178 470)(109 246 179 471)(110 247 180 472)(111 248 166 473)(112 249 167 474)(113 250 168 475)(114 251 169 476)(115 252 170 477)(116 253 171 478)(117 254 172 479)(118 255 173 480)(119 241 174 466)(120 242 175 467)(151 440 382 272)(152 441 383 273)(153 442 384 274)(154 443 385 275)(155 444 386 276)(156 445 387 277)(157 446 388 278)(158 447 389 279)(159 448 390 280)(160 449 376 281)(161 450 377 282)(162 436 378 283)(163 437 379 284)(164 438 380 285)(165 439 381 271)(196 270 394 421)(197 256 395 422)(198 257 396 423)(199 258 397 424)(200 259 398 425)(201 260 399 426)(202 261 400 427)(203 262 401 428)(204 263 402 429)(205 264 403 430)(206 265 404 431)(207 266 405 432)(208 267 391 433)(209 268 392 434)(210 269 393 435)
(1 110 139 180)(2 111 140 166)(3 112 141 167)(4 113 142 168)(5 114 143 169)(6 115 144 170)(7 116 145 171)(8 117 146 172)(9 118 147 173)(10 119 148 174)(11 120 149 175)(12 106 150 176)(13 107 136 177)(14 108 137 178)(15 109 138 179)(16 262 347 428)(17 263 348 429)(18 264 349 430)(19 265 350 431)(20 266 351 432)(21 267 352 433)(22 268 353 434)(23 269 354 435)(24 270 355 421)(25 256 356 422)(26 257 357 423)(27 258 358 424)(28 259 359 425)(29 260 360 426)(30 261 346 427)(31 193 224 91)(32 194 225 92)(33 195 211 93)(34 181 212 94)(35 182 213 95)(36 183 214 96)(37 184 215 97)(38 185 216 98)(39 186 217 99)(40 187 218 100)(41 188 219 101)(42 189 220 102)(43 190 221 103)(44 191 222 104)(45 192 223 105)(46 409 319 465)(47 410 320 451)(48 411 321 452)(49 412 322 453)(50 413 323 454)(51 414 324 455)(52 415 325 456)(53 416 326 457)(54 417 327 458)(55 418 328 459)(56 419 329 460)(57 420 330 461)(58 406 316 462)(59 407 317 463)(60 408 318 464)(61 471 128 246)(62 472 129 247)(63 473 130 248)(64 474 131 249)(65 475 132 250)(66 476 133 251)(67 477 134 252)(68 478 135 253)(69 479 121 254)(70 480 122 255)(71 466 123 241)(72 467 124 242)(73 468 125 243)(74 469 126 244)(75 470 127 245)(76 379 334 163)(77 380 335 164)(78 381 336 165)(79 382 337 151)(80 383 338 152)(81 384 339 153)(82 385 340 154)(83 386 341 155)(84 387 342 156)(85 388 343 157)(86 389 344 158)(87 390 345 159)(88 376 331 160)(89 377 332 161)(90 378 333 162)(196 370 394 292)(197 371 395 293)(198 372 396 294)(199 373 397 295)(200 374 398 296)(201 375 399 297)(202 361 400 298)(203 362 401 299)(204 363 402 300)(205 364 403 286)(206 365 404 287)(207 366 405 288)(208 367 391 289)(209 368 392 290)(210 369 393 291)(226 282 301 450)(227 283 302 436)(228 284 303 437)(229 285 304 438)(230 271 305 439)(231 272 306 440)(232 273 307 441)(233 274 308 442)(234 275 309 443)(235 276 310 444)(236 277 311 445)(237 278 312 446)(238 279 313 447)(239 280 314 448)(240 281 315 449)
G:=sub<Sym(480)| (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), (1,100,307,259)(2,101,308,260)(3,102,309,261)(4,103,310,262)(5,104,311,263)(6,105,312,264)(7,91,313,265)(8,92,314,266)(9,93,315,267)(10,94,301,268)(11,95,302,269)(12,96,303,270)(13,97,304,256)(14,98,305,257)(15,99,306,258)(16,113,221,444)(17,114,222,445)(18,115,223,446)(19,116,224,447)(20,117,225,448)(21,118,211,449)(22,119,212,450)(23,120,213,436)(24,106,214,437)(25,107,215,438)(26,108,216,439)(27,109,217,440)(28,110,218,441)(29,111,219,442)(30,112,220,443)(31,279,350,171)(32,280,351,172)(33,281,352,173)(34,282,353,174)(35,283,354,175)(36,284,355,176)(37,285,356,177)(38,271,357,178)(39,272,358,179)(40,273,359,180)(41,274,360,166)(42,275,346,167)(43,276,347,168)(44,277,348,169)(45,278,349,170)(46,474,298,154)(47,475,299,155)(48,476,300,156)(49,477,286,157)(50,478,287,158)(51,479,288,159)(52,480,289,160)(53,466,290,161)(54,467,291,162)(55,468,292,163)(56,469,293,164)(57,470,294,165)(58,471,295,151)(59,472,296,152)(60,473,297,153)(61,199,337,406)(62,200,338,407)(63,201,339,408)(64,202,340,409)(65,203,341,410)(66,204,342,411)(67,205,343,412)(68,206,344,413)(69,207,345,414)(70,208,331,415)(71,209,332,416)(72,210,333,417)(73,196,334,418)(74,197,335,419)(75,198,336,420)(76,459,125,394)(77,460,126,395)(78,461,127,396)(79,462,128,397)(80,463,129,398)(81,464,130,399)(82,465,131,400)(83,451,132,401)(84,452,133,402)(85,453,134,403)(86,454,135,404)(87,455,121,405)(88,456,122,391)(89,457,123,392)(90,458,124,393)(136,184,229,422)(137,185,230,423)(138,186,231,424)(139,187,232,425)(140,188,233,426)(141,189,234,427)(142,190,235,428)(143,191,236,429)(144,192,237,430)(145,193,238,431)(146,194,239,432)(147,195,240,433)(148,181,226,434)(149,182,227,435)(150,183,228,421)(241,368,377,326)(242,369,378,327)(243,370,379,328)(244,371,380,329)(245,372,381,330)(246,373,382,316)(247,374,383,317)(248,375,384,318)(249,361,385,319)(250,362,386,320)(251,363,387,321)(252,364,388,322)(253,365,389,323)(254,366,390,324)(255,367,376,325), (1,80,139,338)(2,81,140,339)(3,82,141,340)(4,83,142,341)(5,84,143,342)(6,85,144,343)(7,86,145,344)(8,87,146,345)(9,88,147,331)(10,89,148,332)(11,90,149,333)(12,76,150,334)(13,77,136,335)(14,78,137,336)(15,79,138,337)(16,320,347,47)(17,321,348,48)(18,322,349,49)(19,323,350,50)(20,324,351,51)(21,325,352,52)(22,326,353,53)(23,327,354,54)(24,328,355,55)(25,329,356,56)(26,330,357,57)(27,316,358,58)(28,317,359,59)(29,318,360,60)(30,319,346,46)(31,287,224,365)(32,288,225,366)(33,289,211,367)(34,290,212,368)(35,291,213,369)(36,292,214,370)(37,293,215,371)(38,294,216,372)(39,295,217,373)(40,296,218,374)(41,297,219,375)(42,298,220,361)(43,299,221,362)(44,300,222,363)(45,286,223,364)(61,306,128,231)(62,307,129,232)(63,308,130,233)(64,309,131,234)(65,310,132,235)(66,311,133,236)(67,312,134,237)(68,313,135,238)(69,314,121,239)(70,315,122,240)(71,301,123,226)(72,302,124,227)(73,303,125,228)(74,304,126,229)(75,305,127,230)(91,454,193,413)(92,455,194,414)(93,456,195,415)(94,457,181,416)(95,458,182,417)(96,459,183,418)(97,460,184,419)(98,461,185,420)(99,462,186,406)(100,463,187,407)(101,464,188,408)(102,465,189,409)(103,451,190,410)(104,452,191,411)(105,453,192,412)(106,243,176,468)(107,244,177,469)(108,245,178,470)(109,246,179,471)(110,247,180,472)(111,248,166,473)(112,249,167,474)(113,250,168,475)(114,251,169,476)(115,252,170,477)(116,253,171,478)(117,254,172,479)(118,255,173,480)(119,241,174,466)(120,242,175,467)(151,440,382,272)(152,441,383,273)(153,442,384,274)(154,443,385,275)(155,444,386,276)(156,445,387,277)(157,446,388,278)(158,447,389,279)(159,448,390,280)(160,449,376,281)(161,450,377,282)(162,436,378,283)(163,437,379,284)(164,438,380,285)(165,439,381,271)(196,270,394,421)(197,256,395,422)(198,257,396,423)(199,258,397,424)(200,259,398,425)(201,260,399,426)(202,261,400,427)(203,262,401,428)(204,263,402,429)(205,264,403,430)(206,265,404,431)(207,266,405,432)(208,267,391,433)(209,268,392,434)(210,269,393,435), (1,110,139,180)(2,111,140,166)(3,112,141,167)(4,113,142,168)(5,114,143,169)(6,115,144,170)(7,116,145,171)(8,117,146,172)(9,118,147,173)(10,119,148,174)(11,120,149,175)(12,106,150,176)(13,107,136,177)(14,108,137,178)(15,109,138,179)(16,262,347,428)(17,263,348,429)(18,264,349,430)(19,265,350,431)(20,266,351,432)(21,267,352,433)(22,268,353,434)(23,269,354,435)(24,270,355,421)(25,256,356,422)(26,257,357,423)(27,258,358,424)(28,259,359,425)(29,260,360,426)(30,261,346,427)(31,193,224,91)(32,194,225,92)(33,195,211,93)(34,181,212,94)(35,182,213,95)(36,183,214,96)(37,184,215,97)(38,185,216,98)(39,186,217,99)(40,187,218,100)(41,188,219,101)(42,189,220,102)(43,190,221,103)(44,191,222,104)(45,192,223,105)(46,409,319,465)(47,410,320,451)(48,411,321,452)(49,412,322,453)(50,413,323,454)(51,414,324,455)(52,415,325,456)(53,416,326,457)(54,417,327,458)(55,418,328,459)(56,419,329,460)(57,420,330,461)(58,406,316,462)(59,407,317,463)(60,408,318,464)(61,471,128,246)(62,472,129,247)(63,473,130,248)(64,474,131,249)(65,475,132,250)(66,476,133,251)(67,477,134,252)(68,478,135,253)(69,479,121,254)(70,480,122,255)(71,466,123,241)(72,467,124,242)(73,468,125,243)(74,469,126,244)(75,470,127,245)(76,379,334,163)(77,380,335,164)(78,381,336,165)(79,382,337,151)(80,383,338,152)(81,384,339,153)(82,385,340,154)(83,386,341,155)(84,387,342,156)(85,388,343,157)(86,389,344,158)(87,390,345,159)(88,376,331,160)(89,377,332,161)(90,378,333,162)(196,370,394,292)(197,371,395,293)(198,372,396,294)(199,373,397,295)(200,374,398,296)(201,375,399,297)(202,361,400,298)(203,362,401,299)(204,363,402,300)(205,364,403,286)(206,365,404,287)(207,366,405,288)(208,367,391,289)(209,368,392,290)(210,369,393,291)(226,282,301,450)(227,283,302,436)(228,284,303,437)(229,285,304,438)(230,271,305,439)(231,272,306,440)(232,273,307,441)(233,274,308,442)(234,275,309,443)(235,276,310,444)(236,277,311,445)(237,278,312,446)(238,279,313,447)(239,280,314,448)(240,281,315,449)>;
G:=Group( (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), (1,100,307,259)(2,101,308,260)(3,102,309,261)(4,103,310,262)(5,104,311,263)(6,105,312,264)(7,91,313,265)(8,92,314,266)(9,93,315,267)(10,94,301,268)(11,95,302,269)(12,96,303,270)(13,97,304,256)(14,98,305,257)(15,99,306,258)(16,113,221,444)(17,114,222,445)(18,115,223,446)(19,116,224,447)(20,117,225,448)(21,118,211,449)(22,119,212,450)(23,120,213,436)(24,106,214,437)(25,107,215,438)(26,108,216,439)(27,109,217,440)(28,110,218,441)(29,111,219,442)(30,112,220,443)(31,279,350,171)(32,280,351,172)(33,281,352,173)(34,282,353,174)(35,283,354,175)(36,284,355,176)(37,285,356,177)(38,271,357,178)(39,272,358,179)(40,273,359,180)(41,274,360,166)(42,275,346,167)(43,276,347,168)(44,277,348,169)(45,278,349,170)(46,474,298,154)(47,475,299,155)(48,476,300,156)(49,477,286,157)(50,478,287,158)(51,479,288,159)(52,480,289,160)(53,466,290,161)(54,467,291,162)(55,468,292,163)(56,469,293,164)(57,470,294,165)(58,471,295,151)(59,472,296,152)(60,473,297,153)(61,199,337,406)(62,200,338,407)(63,201,339,408)(64,202,340,409)(65,203,341,410)(66,204,342,411)(67,205,343,412)(68,206,344,413)(69,207,345,414)(70,208,331,415)(71,209,332,416)(72,210,333,417)(73,196,334,418)(74,197,335,419)(75,198,336,420)(76,459,125,394)(77,460,126,395)(78,461,127,396)(79,462,128,397)(80,463,129,398)(81,464,130,399)(82,465,131,400)(83,451,132,401)(84,452,133,402)(85,453,134,403)(86,454,135,404)(87,455,121,405)(88,456,122,391)(89,457,123,392)(90,458,124,393)(136,184,229,422)(137,185,230,423)(138,186,231,424)(139,187,232,425)(140,188,233,426)(141,189,234,427)(142,190,235,428)(143,191,236,429)(144,192,237,430)(145,193,238,431)(146,194,239,432)(147,195,240,433)(148,181,226,434)(149,182,227,435)(150,183,228,421)(241,368,377,326)(242,369,378,327)(243,370,379,328)(244,371,380,329)(245,372,381,330)(246,373,382,316)(247,374,383,317)(248,375,384,318)(249,361,385,319)(250,362,386,320)(251,363,387,321)(252,364,388,322)(253,365,389,323)(254,366,390,324)(255,367,376,325), (1,80,139,338)(2,81,140,339)(3,82,141,340)(4,83,142,341)(5,84,143,342)(6,85,144,343)(7,86,145,344)(8,87,146,345)(9,88,147,331)(10,89,148,332)(11,90,149,333)(12,76,150,334)(13,77,136,335)(14,78,137,336)(15,79,138,337)(16,320,347,47)(17,321,348,48)(18,322,349,49)(19,323,350,50)(20,324,351,51)(21,325,352,52)(22,326,353,53)(23,327,354,54)(24,328,355,55)(25,329,356,56)(26,330,357,57)(27,316,358,58)(28,317,359,59)(29,318,360,60)(30,319,346,46)(31,287,224,365)(32,288,225,366)(33,289,211,367)(34,290,212,368)(35,291,213,369)(36,292,214,370)(37,293,215,371)(38,294,216,372)(39,295,217,373)(40,296,218,374)(41,297,219,375)(42,298,220,361)(43,299,221,362)(44,300,222,363)(45,286,223,364)(61,306,128,231)(62,307,129,232)(63,308,130,233)(64,309,131,234)(65,310,132,235)(66,311,133,236)(67,312,134,237)(68,313,135,238)(69,314,121,239)(70,315,122,240)(71,301,123,226)(72,302,124,227)(73,303,125,228)(74,304,126,229)(75,305,127,230)(91,454,193,413)(92,455,194,414)(93,456,195,415)(94,457,181,416)(95,458,182,417)(96,459,183,418)(97,460,184,419)(98,461,185,420)(99,462,186,406)(100,463,187,407)(101,464,188,408)(102,465,189,409)(103,451,190,410)(104,452,191,411)(105,453,192,412)(106,243,176,468)(107,244,177,469)(108,245,178,470)(109,246,179,471)(110,247,180,472)(111,248,166,473)(112,249,167,474)(113,250,168,475)(114,251,169,476)(115,252,170,477)(116,253,171,478)(117,254,172,479)(118,255,173,480)(119,241,174,466)(120,242,175,467)(151,440,382,272)(152,441,383,273)(153,442,384,274)(154,443,385,275)(155,444,386,276)(156,445,387,277)(157,446,388,278)(158,447,389,279)(159,448,390,280)(160,449,376,281)(161,450,377,282)(162,436,378,283)(163,437,379,284)(164,438,380,285)(165,439,381,271)(196,270,394,421)(197,256,395,422)(198,257,396,423)(199,258,397,424)(200,259,398,425)(201,260,399,426)(202,261,400,427)(203,262,401,428)(204,263,402,429)(205,264,403,430)(206,265,404,431)(207,266,405,432)(208,267,391,433)(209,268,392,434)(210,269,393,435), (1,110,139,180)(2,111,140,166)(3,112,141,167)(4,113,142,168)(5,114,143,169)(6,115,144,170)(7,116,145,171)(8,117,146,172)(9,118,147,173)(10,119,148,174)(11,120,149,175)(12,106,150,176)(13,107,136,177)(14,108,137,178)(15,109,138,179)(16,262,347,428)(17,263,348,429)(18,264,349,430)(19,265,350,431)(20,266,351,432)(21,267,352,433)(22,268,353,434)(23,269,354,435)(24,270,355,421)(25,256,356,422)(26,257,357,423)(27,258,358,424)(28,259,359,425)(29,260,360,426)(30,261,346,427)(31,193,224,91)(32,194,225,92)(33,195,211,93)(34,181,212,94)(35,182,213,95)(36,183,214,96)(37,184,215,97)(38,185,216,98)(39,186,217,99)(40,187,218,100)(41,188,219,101)(42,189,220,102)(43,190,221,103)(44,191,222,104)(45,192,223,105)(46,409,319,465)(47,410,320,451)(48,411,321,452)(49,412,322,453)(50,413,323,454)(51,414,324,455)(52,415,325,456)(53,416,326,457)(54,417,327,458)(55,418,328,459)(56,419,329,460)(57,420,330,461)(58,406,316,462)(59,407,317,463)(60,408,318,464)(61,471,128,246)(62,472,129,247)(63,473,130,248)(64,474,131,249)(65,475,132,250)(66,476,133,251)(67,477,134,252)(68,478,135,253)(69,479,121,254)(70,480,122,255)(71,466,123,241)(72,467,124,242)(73,468,125,243)(74,469,126,244)(75,470,127,245)(76,379,334,163)(77,380,335,164)(78,381,336,165)(79,382,337,151)(80,383,338,152)(81,384,339,153)(82,385,340,154)(83,386,341,155)(84,387,342,156)(85,388,343,157)(86,389,344,158)(87,390,345,159)(88,376,331,160)(89,377,332,161)(90,378,333,162)(196,370,394,292)(197,371,395,293)(198,372,396,294)(199,373,397,295)(200,374,398,296)(201,375,399,297)(202,361,400,298)(203,362,401,299)(204,363,402,300)(205,364,403,286)(206,365,404,287)(207,366,405,288)(208,367,391,289)(209,368,392,290)(210,369,393,291)(226,282,301,450)(227,283,302,436)(228,284,303,437)(229,285,304,438)(230,271,305,439)(231,272,306,440)(232,273,307,441)(233,274,308,442)(234,275,309,443)(235,276,310,444)(236,277,311,445)(237,278,312,446)(238,279,313,447)(239,280,314,448)(240,281,315,449) );
G=PermutationGroup([[(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)], [(1,100,307,259),(2,101,308,260),(3,102,309,261),(4,103,310,262),(5,104,311,263),(6,105,312,264),(7,91,313,265),(8,92,314,266),(9,93,315,267),(10,94,301,268),(11,95,302,269),(12,96,303,270),(13,97,304,256),(14,98,305,257),(15,99,306,258),(16,113,221,444),(17,114,222,445),(18,115,223,446),(19,116,224,447),(20,117,225,448),(21,118,211,449),(22,119,212,450),(23,120,213,436),(24,106,214,437),(25,107,215,438),(26,108,216,439),(27,109,217,440),(28,110,218,441),(29,111,219,442),(30,112,220,443),(31,279,350,171),(32,280,351,172),(33,281,352,173),(34,282,353,174),(35,283,354,175),(36,284,355,176),(37,285,356,177),(38,271,357,178),(39,272,358,179),(40,273,359,180),(41,274,360,166),(42,275,346,167),(43,276,347,168),(44,277,348,169),(45,278,349,170),(46,474,298,154),(47,475,299,155),(48,476,300,156),(49,477,286,157),(50,478,287,158),(51,479,288,159),(52,480,289,160),(53,466,290,161),(54,467,291,162),(55,468,292,163),(56,469,293,164),(57,470,294,165),(58,471,295,151),(59,472,296,152),(60,473,297,153),(61,199,337,406),(62,200,338,407),(63,201,339,408),(64,202,340,409),(65,203,341,410),(66,204,342,411),(67,205,343,412),(68,206,344,413),(69,207,345,414),(70,208,331,415),(71,209,332,416),(72,210,333,417),(73,196,334,418),(74,197,335,419),(75,198,336,420),(76,459,125,394),(77,460,126,395),(78,461,127,396),(79,462,128,397),(80,463,129,398),(81,464,130,399),(82,465,131,400),(83,451,132,401),(84,452,133,402),(85,453,134,403),(86,454,135,404),(87,455,121,405),(88,456,122,391),(89,457,123,392),(90,458,124,393),(136,184,229,422),(137,185,230,423),(138,186,231,424),(139,187,232,425),(140,188,233,426),(141,189,234,427),(142,190,235,428),(143,191,236,429),(144,192,237,430),(145,193,238,431),(146,194,239,432),(147,195,240,433),(148,181,226,434),(149,182,227,435),(150,183,228,421),(241,368,377,326),(242,369,378,327),(243,370,379,328),(244,371,380,329),(245,372,381,330),(246,373,382,316),(247,374,383,317),(248,375,384,318),(249,361,385,319),(250,362,386,320),(251,363,387,321),(252,364,388,322),(253,365,389,323),(254,366,390,324),(255,367,376,325)], [(1,80,139,338),(2,81,140,339),(3,82,141,340),(4,83,142,341),(5,84,143,342),(6,85,144,343),(7,86,145,344),(8,87,146,345),(9,88,147,331),(10,89,148,332),(11,90,149,333),(12,76,150,334),(13,77,136,335),(14,78,137,336),(15,79,138,337),(16,320,347,47),(17,321,348,48),(18,322,349,49),(19,323,350,50),(20,324,351,51),(21,325,352,52),(22,326,353,53),(23,327,354,54),(24,328,355,55),(25,329,356,56),(26,330,357,57),(27,316,358,58),(28,317,359,59),(29,318,360,60),(30,319,346,46),(31,287,224,365),(32,288,225,366),(33,289,211,367),(34,290,212,368),(35,291,213,369),(36,292,214,370),(37,293,215,371),(38,294,216,372),(39,295,217,373),(40,296,218,374),(41,297,219,375),(42,298,220,361),(43,299,221,362),(44,300,222,363),(45,286,223,364),(61,306,128,231),(62,307,129,232),(63,308,130,233),(64,309,131,234),(65,310,132,235),(66,311,133,236),(67,312,134,237),(68,313,135,238),(69,314,121,239),(70,315,122,240),(71,301,123,226),(72,302,124,227),(73,303,125,228),(74,304,126,229),(75,305,127,230),(91,454,193,413),(92,455,194,414),(93,456,195,415),(94,457,181,416),(95,458,182,417),(96,459,183,418),(97,460,184,419),(98,461,185,420),(99,462,186,406),(100,463,187,407),(101,464,188,408),(102,465,189,409),(103,451,190,410),(104,452,191,411),(105,453,192,412),(106,243,176,468),(107,244,177,469),(108,245,178,470),(109,246,179,471),(110,247,180,472),(111,248,166,473),(112,249,167,474),(113,250,168,475),(114,251,169,476),(115,252,170,477),(116,253,171,478),(117,254,172,479),(118,255,173,480),(119,241,174,466),(120,242,175,467),(151,440,382,272),(152,441,383,273),(153,442,384,274),(154,443,385,275),(155,444,386,276),(156,445,387,277),(157,446,388,278),(158,447,389,279),(159,448,390,280),(160,449,376,281),(161,450,377,282),(162,436,378,283),(163,437,379,284),(164,438,380,285),(165,439,381,271),(196,270,394,421),(197,256,395,422),(198,257,396,423),(199,258,397,424),(200,259,398,425),(201,260,399,426),(202,261,400,427),(203,262,401,428),(204,263,402,429),(205,264,403,430),(206,265,404,431),(207,266,405,432),(208,267,391,433),(209,268,392,434),(210,269,393,435)], [(1,110,139,180),(2,111,140,166),(3,112,141,167),(4,113,142,168),(5,114,143,169),(6,115,144,170),(7,116,145,171),(8,117,146,172),(9,118,147,173),(10,119,148,174),(11,120,149,175),(12,106,150,176),(13,107,136,177),(14,108,137,178),(15,109,138,179),(16,262,347,428),(17,263,348,429),(18,264,349,430),(19,265,350,431),(20,266,351,432),(21,267,352,433),(22,268,353,434),(23,269,354,435),(24,270,355,421),(25,256,356,422),(26,257,357,423),(27,258,358,424),(28,259,359,425),(29,260,360,426),(30,261,346,427),(31,193,224,91),(32,194,225,92),(33,195,211,93),(34,181,212,94),(35,182,213,95),(36,183,214,96),(37,184,215,97),(38,185,216,98),(39,186,217,99),(40,187,218,100),(41,188,219,101),(42,189,220,102),(43,190,221,103),(44,191,222,104),(45,192,223,105),(46,409,319,465),(47,410,320,451),(48,411,321,452),(49,412,322,453),(50,413,323,454),(51,414,324,455),(52,415,325,456),(53,416,326,457),(54,417,327,458),(55,418,328,459),(56,419,329,460),(57,420,330,461),(58,406,316,462),(59,407,317,463),(60,408,318,464),(61,471,128,246),(62,472,129,247),(63,473,130,248),(64,474,131,249),(65,475,132,250),(66,476,133,251),(67,477,134,252),(68,478,135,253),(69,479,121,254),(70,480,122,255),(71,466,123,241),(72,467,124,242),(73,468,125,243),(74,469,126,244),(75,470,127,245),(76,379,334,163),(77,380,335,164),(78,381,336,165),(79,382,337,151),(80,383,338,152),(81,384,339,153),(82,385,340,154),(83,386,341,155),(84,387,342,156),(85,388,343,157),(86,389,344,158),(87,390,345,159),(88,376,331,160),(89,377,332,161),(90,378,333,162),(196,370,394,292),(197,371,395,293),(198,372,396,294),(199,373,397,295),(200,374,398,296),(201,375,399,297),(202,361,400,298),(203,362,401,299),(204,363,402,300),(205,364,403,286),(206,365,404,287),(207,366,405,288),(208,367,391,289),(209,368,392,290),(210,369,393,291),(226,282,301,450),(227,283,302,436),(228,284,303,437),(229,285,304,438),(230,271,305,439),(231,272,306,440),(232,273,307,441),(233,274,308,442),(234,275,309,443),(235,276,310,444),(236,277,311,445),(237,278,312,446),(238,279,313,447),(239,280,314,448),(240,281,315,449)]])
210 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 10A | ··· | 10L | 12A | ··· | 12L | 12M | ··· | 12T | 15A | ··· | 15H | 20A | ··· | 20X | 20Y | ··· | 20AN | 30A | ··· | 30X | 60A | ··· | 60AV | 60AW | ··· | 60CB |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
210 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||||||||||||
| image | C1 | C2 | C2 | C3 | C5 | C6 | C6 | C10 | C10 | C15 | C30 | C30 | Q8 | C4○D4 | C3×Q8 | C5×Q8 | C3×C4○D4 | C5×C4○D4 | Q8×C15 | C15×C4○D4 |
| kernel | C15×C42.C2 | C4×C60 | C15×C4⋊C4 | C5×C42.C2 | C3×C42.C2 | C4×C20 | C5×C4⋊C4 | C4×C12 | C3×C4⋊C4 | C42.C2 | C42 | C4⋊C4 | C60 | C30 | C20 | C12 | C10 | C6 | C4 | C2 |
| # reps | 1 | 1 | 6 | 2 | 4 | 2 | 12 | 4 | 24 | 8 | 8 | 48 | 2 | 4 | 4 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C15×C42.C2 ►in GL4(𝔽61) generated by
| 47 | 0 | 0 | 0 |
| 0 | 47 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 11 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 0 |
| 0 | 50 | 0 | 0 |
| 11 | 0 | 0 | 0 |
| 0 | 0 | 59 | 19 |
| 0 | 0 | 19 | 2 |
G:=sub<GL(4,GF(61))| [47,0,0,0,0,47,0,0,0,0,15,0,0,0,0,15],[11,0,0,0,0,11,0,0,0,0,0,60,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,60,0,0,1,0],[0,11,0,0,50,0,0,0,0,0,59,19,0,0,19,2] >;
C15×C42.C2 in GAP, Magma, Sage, TeX
C_{15}\times C_4^2.C_2 % in TeX
G:=Group("C15xC4^2.C2"); // GroupNames label
G:=SmallGroup(480,930);
// by ID
G=gap.SmallGroup(480,930);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,840,1709,1688,5126,646]);
// Polycyclic
G:=Group<a,b,c,d|a^15=b^4=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^2,d*c*d^-1=b^2*c>;
// generators/relations