metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.1D15, C30.26C42, C30.34M4(2), C15⋊3C8⋊9C4, (C4×C60).7C2, (C4×C20).7S3, (C4×C12).7D5, C20.89(C4×S3), (C2×C60).32C4, (C2×C4).89D30, C4.19(C4×D15), C12.57(C4×D5), C15⋊11(C8⋊C4), C6.9(C4×Dic5), C60.194(C2×C4), (C2×C20).403D6, (C2×C4).2Dic15, C2.3(C4×Dic15), (C2×C12).5Dic5, (C2×C12).407D10, C5⋊4(C42.S3), (C2×C20).16Dic3, C10.21(C4×Dic3), C3⋊2(C42.D5), C6.4(C4.Dic5), C2.1(C60.7C4), (C2×C60).489C22, C10.9(C4.Dic3), C22.7(C2×Dic15), (C2×C15⋊3C8).7C2, (C2×C30).170(C2×C4), (C2×C6).26(C2×Dic5), (C2×C10).46(C2×Dic3), SmallGroup(480,163)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.D15
G = < a,b,c,d | a4=b4=c15=1, d2=b, ab=ba, ac=ca, dad-1=ab2, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 228 in 80 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, C10, C10, C12, C12, C2×C6, C15, C42, C2×C8, C20, C20, C2×C10, C3⋊C8, C2×C12, C2×C12, C30, C30, C8⋊C4, C5⋊2C8, C2×C20, C2×C20, C2×C3⋊C8, C4×C12, C60, C60, C2×C30, C2×C5⋊2C8, C4×C20, C42.S3, C15⋊3C8, C2×C60, C2×C60, C42.D5, C2×C15⋊3C8, C4×C60, C42.D15
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, Dic3, D6, C42, M4(2), Dic5, D10, C4×S3, C2×Dic3, D15, C8⋊C4, C4×D5, C2×Dic5, C4.Dic3, C4×Dic3, Dic15, D30, C4.Dic5, C4×Dic5, C42.S3, C4×D15, C2×Dic15, C42.D5, C60.7C4, C4×Dic15, C42.D15
►Regular action on 480 points
Generators in S480
(1 151 39 133)(2 152 40 134)(3 153 41 135)(4 154 42 121)(5 155 43 122)(6 156 44 123)(7 157 45 124)(8 158 31 125)(9 159 32 126)(10 160 33 127)(11 161 34 128)(12 162 35 129)(13 163 36 130)(14 164 37 131)(15 165 38 132)(16 177 50 143)(17 178 51 144)(18 179 52 145)(19 180 53 146)(20 166 54 147)(21 167 55 148)(22 168 56 149)(23 169 57 150)(24 170 58 136)(25 171 59 137)(26 172 60 138)(27 173 46 139)(28 174 47 140)(29 175 48 141)(30 176 49 142)(61 225 102 188)(62 211 103 189)(63 212 104 190)(64 213 105 191)(65 214 91 192)(66 215 92 193)(67 216 93 194)(68 217 94 195)(69 218 95 181)(70 219 96 182)(71 220 97 183)(72 221 98 184)(73 222 99 185)(74 223 100 186)(75 224 101 187)(76 233 106 196)(77 234 107 197)(78 235 108 198)(79 236 109 199)(80 237 110 200)(81 238 111 201)(82 239 112 202)(83 240 113 203)(84 226 114 204)(85 227 115 205)(86 228 116 206)(87 229 117 207)(88 230 118 208)(89 231 119 209)(90 232 120 210)(241 410 282 382)(242 411 283 383)(243 412 284 384)(244 413 285 385)(245 414 271 386)(246 415 272 387)(247 416 273 388)(248 417 274 389)(249 418 275 390)(250 419 276 376)(251 420 277 377)(252 406 278 378)(253 407 279 379)(254 408 280 380)(255 409 281 381)(256 396 292 365)(257 397 293 366)(258 398 294 367)(259 399 295 368)(260 400 296 369)(261 401 297 370)(262 402 298 371)(263 403 299 372)(264 404 300 373)(265 405 286 374)(266 391 287 375)(267 392 288 361)(268 393 289 362)(269 394 290 363)(270 395 291 364)(301 471 331 443)(302 472 332 444)(303 473 333 445)(304 474 334 446)(305 475 335 447)(306 476 336 448)(307 477 337 449)(308 478 338 450)(309 479 339 436)(310 480 340 437)(311 466 341 438)(312 467 342 439)(313 468 343 440)(314 469 344 441)(315 470 345 442)(316 465 354 435)(317 451 355 421)(318 452 356 422)(319 453 357 423)(320 454 358 424)(321 455 359 425)(322 456 360 426)(323 457 346 427)(324 458 347 428)(325 459 348 429)(326 460 349 430)(327 461 350 431)(328 462 351 432)(329 463 352 433)(330 464 353 434)
(1 84 28 62)(2 85 29 63)(3 86 30 64)(4 87 16 65)(5 88 17 66)(6 89 18 67)(7 90 19 68)(8 76 20 69)(9 77 21 70)(10 78 22 71)(11 79 23 72)(12 80 24 73)(13 81 25 74)(14 82 26 75)(15 83 27 61)(31 106 54 95)(32 107 55 96)(33 108 56 97)(34 109 57 98)(35 110 58 99)(36 111 59 100)(37 112 60 101)(38 113 46 102)(39 114 47 103)(40 115 48 104)(41 116 49 105)(42 117 50 91)(43 118 51 92)(44 119 52 93)(45 120 53 94)(121 207 143 192)(122 208 144 193)(123 209 145 194)(124 210 146 195)(125 196 147 181)(126 197 148 182)(127 198 149 183)(128 199 150 184)(129 200 136 185)(130 201 137 186)(131 202 138 187)(132 203 139 188)(133 204 140 189)(134 205 141 190)(135 206 142 191)(151 226 174 211)(152 227 175 212)(153 228 176 213)(154 229 177 214)(155 230 178 215)(156 231 179 216)(157 232 180 217)(158 233 166 218)(159 234 167 219)(160 235 168 220)(161 236 169 221)(162 237 170 222)(163 238 171 223)(164 239 172 224)(165 240 173 225)(241 321 262 303)(242 322 263 304)(243 323 264 305)(244 324 265 306)(245 325 266 307)(246 326 267 308)(247 327 268 309)(248 328 269 310)(249 329 270 311)(250 330 256 312)(251 316 257 313)(252 317 258 314)(253 318 259 315)(254 319 260 301)(255 320 261 302)(271 348 287 337)(272 349 288 338)(273 350 289 339)(274 351 290 340)(275 352 291 341)(276 353 292 342)(277 354 293 343)(278 355 294 344)(279 356 295 345)(280 357 296 331)(281 358 297 332)(282 359 298 333)(283 360 299 334)(284 346 300 335)(285 347 286 336)(361 450 387 430)(362 436 388 431)(363 437 389 432)(364 438 390 433)(365 439 376 434)(366 440 377 435)(367 441 378 421)(368 442 379 422)(369 443 380 423)(370 444 381 424)(371 445 382 425)(372 446 383 426)(373 447 384 427)(374 448 385 428)(375 449 386 429)(391 477 414 459)(392 478 415 460)(393 479 416 461)(394 480 417 462)(395 466 418 463)(396 467 419 464)(397 468 420 465)(398 469 406 451)(399 470 407 452)(400 471 408 453)(401 472 409 454)(402 473 410 455)(403 474 411 456)(404 475 412 457)(405 476 413 458)
(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)
(1 328 84 269 28 310 62 248)(2 327 85 268 29 309 63 247)(3 326 86 267 30 308 64 246)(4 325 87 266 16 307 65 245)(5 324 88 265 17 306 66 244)(6 323 89 264 18 305 67 243)(7 322 90 263 19 304 68 242)(8 321 76 262 20 303 69 241)(9 320 77 261 21 302 70 255)(10 319 78 260 22 301 71 254)(11 318 79 259 23 315 72 253)(12 317 80 258 24 314 73 252)(13 316 81 257 25 313 74 251)(14 330 82 256 26 312 75 250)(15 329 83 270 27 311 61 249)(31 359 106 298 54 333 95 282)(32 358 107 297 55 332 96 281)(33 357 108 296 56 331 97 280)(34 356 109 295 57 345 98 279)(35 355 110 294 58 344 99 278)(36 354 111 293 59 343 100 277)(37 353 112 292 60 342 101 276)(38 352 113 291 46 341 102 275)(39 351 114 290 47 340 103 274)(40 350 115 289 48 339 104 273)(41 349 116 288 49 338 105 272)(42 348 117 287 50 337 91 271)(43 347 118 286 51 336 92 285)(44 346 119 300 52 335 93 284)(45 360 120 299 53 334 94 283)(121 449 207 386 143 429 192 375)(122 448 208 385 144 428 193 374)(123 447 209 384 145 427 194 373)(124 446 210 383 146 426 195 372)(125 445 196 382 147 425 181 371)(126 444 197 381 148 424 182 370)(127 443 198 380 149 423 183 369)(128 442 199 379 150 422 184 368)(129 441 200 378 136 421 185 367)(130 440 201 377 137 435 186 366)(131 439 202 376 138 434 187 365)(132 438 203 390 139 433 188 364)(133 437 204 389 140 432 189 363)(134 436 205 388 141 431 190 362)(135 450 206 387 142 430 191 361)(151 480 226 417 174 462 211 394)(152 479 227 416 175 461 212 393)(153 478 228 415 176 460 213 392)(154 477 229 414 177 459 214 391)(155 476 230 413 178 458 215 405)(156 475 231 412 179 457 216 404)(157 474 232 411 180 456 217 403)(158 473 233 410 166 455 218 402)(159 472 234 409 167 454 219 401)(160 471 235 408 168 453 220 400)(161 470 236 407 169 452 221 399)(162 469 237 406 170 451 222 398)(163 468 238 420 171 465 223 397)(164 467 239 419 172 464 224 396)(165 466 240 418 173 463 225 395)
G:=sub<Sym(480)| (1,151,39,133)(2,152,40,134)(3,153,41,135)(4,154,42,121)(5,155,43,122)(6,156,44,123)(7,157,45,124)(8,158,31,125)(9,159,32,126)(10,160,33,127)(11,161,34,128)(12,162,35,129)(13,163,36,130)(14,164,37,131)(15,165,38,132)(16,177,50,143)(17,178,51,144)(18,179,52,145)(19,180,53,146)(20,166,54,147)(21,167,55,148)(22,168,56,149)(23,169,57,150)(24,170,58,136)(25,171,59,137)(26,172,60,138)(27,173,46,139)(28,174,47,140)(29,175,48,141)(30,176,49,142)(61,225,102,188)(62,211,103,189)(63,212,104,190)(64,213,105,191)(65,214,91,192)(66,215,92,193)(67,216,93,194)(68,217,94,195)(69,218,95,181)(70,219,96,182)(71,220,97,183)(72,221,98,184)(73,222,99,185)(74,223,100,186)(75,224,101,187)(76,233,106,196)(77,234,107,197)(78,235,108,198)(79,236,109,199)(80,237,110,200)(81,238,111,201)(82,239,112,202)(83,240,113,203)(84,226,114,204)(85,227,115,205)(86,228,116,206)(87,229,117,207)(88,230,118,208)(89,231,119,209)(90,232,120,210)(241,410,282,382)(242,411,283,383)(243,412,284,384)(244,413,285,385)(245,414,271,386)(246,415,272,387)(247,416,273,388)(248,417,274,389)(249,418,275,390)(250,419,276,376)(251,420,277,377)(252,406,278,378)(253,407,279,379)(254,408,280,380)(255,409,281,381)(256,396,292,365)(257,397,293,366)(258,398,294,367)(259,399,295,368)(260,400,296,369)(261,401,297,370)(262,402,298,371)(263,403,299,372)(264,404,300,373)(265,405,286,374)(266,391,287,375)(267,392,288,361)(268,393,289,362)(269,394,290,363)(270,395,291,364)(301,471,331,443)(302,472,332,444)(303,473,333,445)(304,474,334,446)(305,475,335,447)(306,476,336,448)(307,477,337,449)(308,478,338,450)(309,479,339,436)(310,480,340,437)(311,466,341,438)(312,467,342,439)(313,468,343,440)(314,469,344,441)(315,470,345,442)(316,465,354,435)(317,451,355,421)(318,452,356,422)(319,453,357,423)(320,454,358,424)(321,455,359,425)(322,456,360,426)(323,457,346,427)(324,458,347,428)(325,459,348,429)(326,460,349,430)(327,461,350,431)(328,462,351,432)(329,463,352,433)(330,464,353,434), (1,84,28,62)(2,85,29,63)(3,86,30,64)(4,87,16,65)(5,88,17,66)(6,89,18,67)(7,90,19,68)(8,76,20,69)(9,77,21,70)(10,78,22,71)(11,79,23,72)(12,80,24,73)(13,81,25,74)(14,82,26,75)(15,83,27,61)(31,106,54,95)(32,107,55,96)(33,108,56,97)(34,109,57,98)(35,110,58,99)(36,111,59,100)(37,112,60,101)(38,113,46,102)(39,114,47,103)(40,115,48,104)(41,116,49,105)(42,117,50,91)(43,118,51,92)(44,119,52,93)(45,120,53,94)(121,207,143,192)(122,208,144,193)(123,209,145,194)(124,210,146,195)(125,196,147,181)(126,197,148,182)(127,198,149,183)(128,199,150,184)(129,200,136,185)(130,201,137,186)(131,202,138,187)(132,203,139,188)(133,204,140,189)(134,205,141,190)(135,206,142,191)(151,226,174,211)(152,227,175,212)(153,228,176,213)(154,229,177,214)(155,230,178,215)(156,231,179,216)(157,232,180,217)(158,233,166,218)(159,234,167,219)(160,235,168,220)(161,236,169,221)(162,237,170,222)(163,238,171,223)(164,239,172,224)(165,240,173,225)(241,321,262,303)(242,322,263,304)(243,323,264,305)(244,324,265,306)(245,325,266,307)(246,326,267,308)(247,327,268,309)(248,328,269,310)(249,329,270,311)(250,330,256,312)(251,316,257,313)(252,317,258,314)(253,318,259,315)(254,319,260,301)(255,320,261,302)(271,348,287,337)(272,349,288,338)(273,350,289,339)(274,351,290,340)(275,352,291,341)(276,353,292,342)(277,354,293,343)(278,355,294,344)(279,356,295,345)(280,357,296,331)(281,358,297,332)(282,359,298,333)(283,360,299,334)(284,346,300,335)(285,347,286,336)(361,450,387,430)(362,436,388,431)(363,437,389,432)(364,438,390,433)(365,439,376,434)(366,440,377,435)(367,441,378,421)(368,442,379,422)(369,443,380,423)(370,444,381,424)(371,445,382,425)(372,446,383,426)(373,447,384,427)(374,448,385,428)(375,449,386,429)(391,477,414,459)(392,478,415,460)(393,479,416,461)(394,480,417,462)(395,466,418,463)(396,467,419,464)(397,468,420,465)(398,469,406,451)(399,470,407,452)(400,471,408,453)(401,472,409,454)(402,473,410,455)(403,474,411,456)(404,475,412,457)(405,476,413,458), (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), (1,328,84,269,28,310,62,248)(2,327,85,268,29,309,63,247)(3,326,86,267,30,308,64,246)(4,325,87,266,16,307,65,245)(5,324,88,265,17,306,66,244)(6,323,89,264,18,305,67,243)(7,322,90,263,19,304,68,242)(8,321,76,262,20,303,69,241)(9,320,77,261,21,302,70,255)(10,319,78,260,22,301,71,254)(11,318,79,259,23,315,72,253)(12,317,80,258,24,314,73,252)(13,316,81,257,25,313,74,251)(14,330,82,256,26,312,75,250)(15,329,83,270,27,311,61,249)(31,359,106,298,54,333,95,282)(32,358,107,297,55,332,96,281)(33,357,108,296,56,331,97,280)(34,356,109,295,57,345,98,279)(35,355,110,294,58,344,99,278)(36,354,111,293,59,343,100,277)(37,353,112,292,60,342,101,276)(38,352,113,291,46,341,102,275)(39,351,114,290,47,340,103,274)(40,350,115,289,48,339,104,273)(41,349,116,288,49,338,105,272)(42,348,117,287,50,337,91,271)(43,347,118,286,51,336,92,285)(44,346,119,300,52,335,93,284)(45,360,120,299,53,334,94,283)(121,449,207,386,143,429,192,375)(122,448,208,385,144,428,193,374)(123,447,209,384,145,427,194,373)(124,446,210,383,146,426,195,372)(125,445,196,382,147,425,181,371)(126,444,197,381,148,424,182,370)(127,443,198,380,149,423,183,369)(128,442,199,379,150,422,184,368)(129,441,200,378,136,421,185,367)(130,440,201,377,137,435,186,366)(131,439,202,376,138,434,187,365)(132,438,203,390,139,433,188,364)(133,437,204,389,140,432,189,363)(134,436,205,388,141,431,190,362)(135,450,206,387,142,430,191,361)(151,480,226,417,174,462,211,394)(152,479,227,416,175,461,212,393)(153,478,228,415,176,460,213,392)(154,477,229,414,177,459,214,391)(155,476,230,413,178,458,215,405)(156,475,231,412,179,457,216,404)(157,474,232,411,180,456,217,403)(158,473,233,410,166,455,218,402)(159,472,234,409,167,454,219,401)(160,471,235,408,168,453,220,400)(161,470,236,407,169,452,221,399)(162,469,237,406,170,451,222,398)(163,468,238,420,171,465,223,397)(164,467,239,419,172,464,224,396)(165,466,240,418,173,463,225,395)>;
G:=Group( (1,151,39,133)(2,152,40,134)(3,153,41,135)(4,154,42,121)(5,155,43,122)(6,156,44,123)(7,157,45,124)(8,158,31,125)(9,159,32,126)(10,160,33,127)(11,161,34,128)(12,162,35,129)(13,163,36,130)(14,164,37,131)(15,165,38,132)(16,177,50,143)(17,178,51,144)(18,179,52,145)(19,180,53,146)(20,166,54,147)(21,167,55,148)(22,168,56,149)(23,169,57,150)(24,170,58,136)(25,171,59,137)(26,172,60,138)(27,173,46,139)(28,174,47,140)(29,175,48,141)(30,176,49,142)(61,225,102,188)(62,211,103,189)(63,212,104,190)(64,213,105,191)(65,214,91,192)(66,215,92,193)(67,216,93,194)(68,217,94,195)(69,218,95,181)(70,219,96,182)(71,220,97,183)(72,221,98,184)(73,222,99,185)(74,223,100,186)(75,224,101,187)(76,233,106,196)(77,234,107,197)(78,235,108,198)(79,236,109,199)(80,237,110,200)(81,238,111,201)(82,239,112,202)(83,240,113,203)(84,226,114,204)(85,227,115,205)(86,228,116,206)(87,229,117,207)(88,230,118,208)(89,231,119,209)(90,232,120,210)(241,410,282,382)(242,411,283,383)(243,412,284,384)(244,413,285,385)(245,414,271,386)(246,415,272,387)(247,416,273,388)(248,417,274,389)(249,418,275,390)(250,419,276,376)(251,420,277,377)(252,406,278,378)(253,407,279,379)(254,408,280,380)(255,409,281,381)(256,396,292,365)(257,397,293,366)(258,398,294,367)(259,399,295,368)(260,400,296,369)(261,401,297,370)(262,402,298,371)(263,403,299,372)(264,404,300,373)(265,405,286,374)(266,391,287,375)(267,392,288,361)(268,393,289,362)(269,394,290,363)(270,395,291,364)(301,471,331,443)(302,472,332,444)(303,473,333,445)(304,474,334,446)(305,475,335,447)(306,476,336,448)(307,477,337,449)(308,478,338,450)(309,479,339,436)(310,480,340,437)(311,466,341,438)(312,467,342,439)(313,468,343,440)(314,469,344,441)(315,470,345,442)(316,465,354,435)(317,451,355,421)(318,452,356,422)(319,453,357,423)(320,454,358,424)(321,455,359,425)(322,456,360,426)(323,457,346,427)(324,458,347,428)(325,459,348,429)(326,460,349,430)(327,461,350,431)(328,462,351,432)(329,463,352,433)(330,464,353,434), (1,84,28,62)(2,85,29,63)(3,86,30,64)(4,87,16,65)(5,88,17,66)(6,89,18,67)(7,90,19,68)(8,76,20,69)(9,77,21,70)(10,78,22,71)(11,79,23,72)(12,80,24,73)(13,81,25,74)(14,82,26,75)(15,83,27,61)(31,106,54,95)(32,107,55,96)(33,108,56,97)(34,109,57,98)(35,110,58,99)(36,111,59,100)(37,112,60,101)(38,113,46,102)(39,114,47,103)(40,115,48,104)(41,116,49,105)(42,117,50,91)(43,118,51,92)(44,119,52,93)(45,120,53,94)(121,207,143,192)(122,208,144,193)(123,209,145,194)(124,210,146,195)(125,196,147,181)(126,197,148,182)(127,198,149,183)(128,199,150,184)(129,200,136,185)(130,201,137,186)(131,202,138,187)(132,203,139,188)(133,204,140,189)(134,205,141,190)(135,206,142,191)(151,226,174,211)(152,227,175,212)(153,228,176,213)(154,229,177,214)(155,230,178,215)(156,231,179,216)(157,232,180,217)(158,233,166,218)(159,234,167,219)(160,235,168,220)(161,236,169,221)(162,237,170,222)(163,238,171,223)(164,239,172,224)(165,240,173,225)(241,321,262,303)(242,322,263,304)(243,323,264,305)(244,324,265,306)(245,325,266,307)(246,326,267,308)(247,327,268,309)(248,328,269,310)(249,329,270,311)(250,330,256,312)(251,316,257,313)(252,317,258,314)(253,318,259,315)(254,319,260,301)(255,320,261,302)(271,348,287,337)(272,349,288,338)(273,350,289,339)(274,351,290,340)(275,352,291,341)(276,353,292,342)(277,354,293,343)(278,355,294,344)(279,356,295,345)(280,357,296,331)(281,358,297,332)(282,359,298,333)(283,360,299,334)(284,346,300,335)(285,347,286,336)(361,450,387,430)(362,436,388,431)(363,437,389,432)(364,438,390,433)(365,439,376,434)(366,440,377,435)(367,441,378,421)(368,442,379,422)(369,443,380,423)(370,444,381,424)(371,445,382,425)(372,446,383,426)(373,447,384,427)(374,448,385,428)(375,449,386,429)(391,477,414,459)(392,478,415,460)(393,479,416,461)(394,480,417,462)(395,466,418,463)(396,467,419,464)(397,468,420,465)(398,469,406,451)(399,470,407,452)(400,471,408,453)(401,472,409,454)(402,473,410,455)(403,474,411,456)(404,475,412,457)(405,476,413,458), (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), (1,328,84,269,28,310,62,248)(2,327,85,268,29,309,63,247)(3,326,86,267,30,308,64,246)(4,325,87,266,16,307,65,245)(5,324,88,265,17,306,66,244)(6,323,89,264,18,305,67,243)(7,322,90,263,19,304,68,242)(8,321,76,262,20,303,69,241)(9,320,77,261,21,302,70,255)(10,319,78,260,22,301,71,254)(11,318,79,259,23,315,72,253)(12,317,80,258,24,314,73,252)(13,316,81,257,25,313,74,251)(14,330,82,256,26,312,75,250)(15,329,83,270,27,311,61,249)(31,359,106,298,54,333,95,282)(32,358,107,297,55,332,96,281)(33,357,108,296,56,331,97,280)(34,356,109,295,57,345,98,279)(35,355,110,294,58,344,99,278)(36,354,111,293,59,343,100,277)(37,353,112,292,60,342,101,276)(38,352,113,291,46,341,102,275)(39,351,114,290,47,340,103,274)(40,350,115,289,48,339,104,273)(41,349,116,288,49,338,105,272)(42,348,117,287,50,337,91,271)(43,347,118,286,51,336,92,285)(44,346,119,300,52,335,93,284)(45,360,120,299,53,334,94,283)(121,449,207,386,143,429,192,375)(122,448,208,385,144,428,193,374)(123,447,209,384,145,427,194,373)(124,446,210,383,146,426,195,372)(125,445,196,382,147,425,181,371)(126,444,197,381,148,424,182,370)(127,443,198,380,149,423,183,369)(128,442,199,379,150,422,184,368)(129,441,200,378,136,421,185,367)(130,440,201,377,137,435,186,366)(131,439,202,376,138,434,187,365)(132,438,203,390,139,433,188,364)(133,437,204,389,140,432,189,363)(134,436,205,388,141,431,190,362)(135,450,206,387,142,430,191,361)(151,480,226,417,174,462,211,394)(152,479,227,416,175,461,212,393)(153,478,228,415,176,460,213,392)(154,477,229,414,177,459,214,391)(155,476,230,413,178,458,215,405)(156,475,231,412,179,457,216,404)(157,474,232,411,180,456,217,403)(158,473,233,410,166,455,218,402)(159,472,234,409,167,454,219,401)(160,471,235,408,168,453,220,400)(161,470,236,407,169,452,221,399)(162,469,237,406,170,451,222,398)(163,468,238,420,171,465,223,397)(164,467,239,419,172,464,224,396)(165,466,240,418,173,463,225,395) );
G=PermutationGroup([[(1,151,39,133),(2,152,40,134),(3,153,41,135),(4,154,42,121),(5,155,43,122),(6,156,44,123),(7,157,45,124),(8,158,31,125),(9,159,32,126),(10,160,33,127),(11,161,34,128),(12,162,35,129),(13,163,36,130),(14,164,37,131),(15,165,38,132),(16,177,50,143),(17,178,51,144),(18,179,52,145),(19,180,53,146),(20,166,54,147),(21,167,55,148),(22,168,56,149),(23,169,57,150),(24,170,58,136),(25,171,59,137),(26,172,60,138),(27,173,46,139),(28,174,47,140),(29,175,48,141),(30,176,49,142),(61,225,102,188),(62,211,103,189),(63,212,104,190),(64,213,105,191),(65,214,91,192),(66,215,92,193),(67,216,93,194),(68,217,94,195),(69,218,95,181),(70,219,96,182),(71,220,97,183),(72,221,98,184),(73,222,99,185),(74,223,100,186),(75,224,101,187),(76,233,106,196),(77,234,107,197),(78,235,108,198),(79,236,109,199),(80,237,110,200),(81,238,111,201),(82,239,112,202),(83,240,113,203),(84,226,114,204),(85,227,115,205),(86,228,116,206),(87,229,117,207),(88,230,118,208),(89,231,119,209),(90,232,120,210),(241,410,282,382),(242,411,283,383),(243,412,284,384),(244,413,285,385),(245,414,271,386),(246,415,272,387),(247,416,273,388),(248,417,274,389),(249,418,275,390),(250,419,276,376),(251,420,277,377),(252,406,278,378),(253,407,279,379),(254,408,280,380),(255,409,281,381),(256,396,292,365),(257,397,293,366),(258,398,294,367),(259,399,295,368),(260,400,296,369),(261,401,297,370),(262,402,298,371),(263,403,299,372),(264,404,300,373),(265,405,286,374),(266,391,287,375),(267,392,288,361),(268,393,289,362),(269,394,290,363),(270,395,291,364),(301,471,331,443),(302,472,332,444),(303,473,333,445),(304,474,334,446),(305,475,335,447),(306,476,336,448),(307,477,337,449),(308,478,338,450),(309,479,339,436),(310,480,340,437),(311,466,341,438),(312,467,342,439),(313,468,343,440),(314,469,344,441),(315,470,345,442),(316,465,354,435),(317,451,355,421),(318,452,356,422),(319,453,357,423),(320,454,358,424),(321,455,359,425),(322,456,360,426),(323,457,346,427),(324,458,347,428),(325,459,348,429),(326,460,349,430),(327,461,350,431),(328,462,351,432),(329,463,352,433),(330,464,353,434)], [(1,84,28,62),(2,85,29,63),(3,86,30,64),(4,87,16,65),(5,88,17,66),(6,89,18,67),(7,90,19,68),(8,76,20,69),(9,77,21,70),(10,78,22,71),(11,79,23,72),(12,80,24,73),(13,81,25,74),(14,82,26,75),(15,83,27,61),(31,106,54,95),(32,107,55,96),(33,108,56,97),(34,109,57,98),(35,110,58,99),(36,111,59,100),(37,112,60,101),(38,113,46,102),(39,114,47,103),(40,115,48,104),(41,116,49,105),(42,117,50,91),(43,118,51,92),(44,119,52,93),(45,120,53,94),(121,207,143,192),(122,208,144,193),(123,209,145,194),(124,210,146,195),(125,196,147,181),(126,197,148,182),(127,198,149,183),(128,199,150,184),(129,200,136,185),(130,201,137,186),(131,202,138,187),(132,203,139,188),(133,204,140,189),(134,205,141,190),(135,206,142,191),(151,226,174,211),(152,227,175,212),(153,228,176,213),(154,229,177,214),(155,230,178,215),(156,231,179,216),(157,232,180,217),(158,233,166,218),(159,234,167,219),(160,235,168,220),(161,236,169,221),(162,237,170,222),(163,238,171,223),(164,239,172,224),(165,240,173,225),(241,321,262,303),(242,322,263,304),(243,323,264,305),(244,324,265,306),(245,325,266,307),(246,326,267,308),(247,327,268,309),(248,328,269,310),(249,329,270,311),(250,330,256,312),(251,316,257,313),(252,317,258,314),(253,318,259,315),(254,319,260,301),(255,320,261,302),(271,348,287,337),(272,349,288,338),(273,350,289,339),(274,351,290,340),(275,352,291,341),(276,353,292,342),(277,354,293,343),(278,355,294,344),(279,356,295,345),(280,357,296,331),(281,358,297,332),(282,359,298,333),(283,360,299,334),(284,346,300,335),(285,347,286,336),(361,450,387,430),(362,436,388,431),(363,437,389,432),(364,438,390,433),(365,439,376,434),(366,440,377,435),(367,441,378,421),(368,442,379,422),(369,443,380,423),(370,444,381,424),(371,445,382,425),(372,446,383,426),(373,447,384,427),(374,448,385,428),(375,449,386,429),(391,477,414,459),(392,478,415,460),(393,479,416,461),(394,480,417,462),(395,466,418,463),(396,467,419,464),(397,468,420,465),(398,469,406,451),(399,470,407,452),(400,471,408,453),(401,472,409,454),(402,473,410,455),(403,474,411,456),(404,475,412,457),(405,476,413,458)], [(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)], [(1,328,84,269,28,310,62,248),(2,327,85,268,29,309,63,247),(3,326,86,267,30,308,64,246),(4,325,87,266,16,307,65,245),(5,324,88,265,17,306,66,244),(6,323,89,264,18,305,67,243),(7,322,90,263,19,304,68,242),(8,321,76,262,20,303,69,241),(9,320,77,261,21,302,70,255),(10,319,78,260,22,301,71,254),(11,318,79,259,23,315,72,253),(12,317,80,258,24,314,73,252),(13,316,81,257,25,313,74,251),(14,330,82,256,26,312,75,250),(15,329,83,270,27,311,61,249),(31,359,106,298,54,333,95,282),(32,358,107,297,55,332,96,281),(33,357,108,296,56,331,97,280),(34,356,109,295,57,345,98,279),(35,355,110,294,58,344,99,278),(36,354,111,293,59,343,100,277),(37,353,112,292,60,342,101,276),(38,352,113,291,46,341,102,275),(39,351,114,290,47,340,103,274),(40,350,115,289,48,339,104,273),(41,349,116,288,49,338,105,272),(42,348,117,287,50,337,91,271),(43,347,118,286,51,336,92,285),(44,346,119,300,52,335,93,284),(45,360,120,299,53,334,94,283),(121,449,207,386,143,429,192,375),(122,448,208,385,144,428,193,374),(123,447,209,384,145,427,194,373),(124,446,210,383,146,426,195,372),(125,445,196,382,147,425,181,371),(126,444,197,381,148,424,182,370),(127,443,198,380,149,423,183,369),(128,442,199,379,150,422,184,368),(129,441,200,378,136,421,185,367),(130,440,201,377,137,435,186,366),(131,439,202,376,138,434,187,365),(132,438,203,390,139,433,188,364),(133,437,204,389,140,432,189,363),(134,436,205,388,141,431,190,362),(135,450,206,387,142,430,191,361),(151,480,226,417,174,462,211,394),(152,479,227,416,175,461,212,393),(153,478,228,415,176,460,213,392),(154,477,229,414,177,459,214,391),(155,476,230,413,178,458,215,405),(156,475,231,412,179,457,216,404),(157,474,232,411,180,456,217,403),(158,473,233,410,166,455,218,402),(159,472,234,409,167,454,219,401),(160,471,235,408,168,453,220,400),(161,470,236,407,169,452,221,399),(162,469,237,406,170,451,222,398),(163,468,238,420,171,465,223,397),(164,467,239,419,172,464,224,396),(165,466,240,418,173,463,225,395)]])
132 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 8A | ··· | 8H | 10A | ··· | 10F | 12A | ··· | 12L | 15A | 15B | 15C | 15D | 20A | ··· | 20X | 30A | ··· | 30L | 60A | ··· | 60AV |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | ··· | 8 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 30 | ··· | 30 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | + | - | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | D5 | Dic3 | D6 | M4(2) | Dic5 | D10 | C4×S3 | D15 | C4×D5 | C4.Dic3 | Dic15 | D30 | C4.Dic5 | C4×D15 | C60.7C4 |
| kernel | C42.D15 | C2×C15⋊3C8 | C4×C60 | C15⋊3C8 | C2×C60 | C4×C20 | C4×C12 | C2×C20 | C2×C20 | C30 | C2×C12 | C2×C12 | C20 | C42 | C12 | C10 | C2×C4 | C2×C4 | C6 | C4 | C2 |
| # reps | 1 | 2 | 1 | 8 | 4 | 1 | 2 | 2 | 1 | 4 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 4 | 16 | 16 | 32 |
Matrix representation of C42.D15 ►in GL5(𝔽241)
| 177 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 165 | 103 |
| 0 | 0 | 0 | 89 | 76 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 240 | 1 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 51 | 189 |
| 0 | 0 | 0 | 51 | 0 |
| 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 201 | 172 |
| 0 | 0 | 0 | 148 | 40 |
G:=sub<GL(5,GF(241))| [177,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,165,89,0,0,0,103,76],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,64,0,0,0,0,0,64],[1,0,0,0,0,0,240,240,0,0,0,1,0,0,0,0,0,0,51,51,0,0,0,189,0],[240,0,0,0,0,0,0,240,0,0,0,240,0,0,0,0,0,0,201,148,0,0,0,172,40] >;
C42.D15 in GAP, Magma, Sage, TeX
C_4^2.D_{15} % in TeX
G:=Group("C4^2.D15"); // GroupNames label
G:=SmallGroup(480,163);
// by ID
G=gap.SmallGroup(480,163);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,64,100,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^15=1,d^2=b,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations