direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C15×C8⋊C4, C8⋊3C60, C24⋊7C20, C120⋊23C4, C40⋊11C12, C42.1C30, C30.41C42, C30.43M4(2), C6.7(C4×C20), C2.2(C4×C60), (C4×C60).1C2, (C2×C8).7C30, (C4×C20).1C6, (C2×C4).2C60, (C2×C40).17C6, (C4×C12).1C10, C4.11(C2×C60), (C2×C60).35C4, (C2×C12).6C20, (C2×C120).37C2, (C2×C24).17C10, (C2×C20).17C12, C10.12(C4×C12), C12.48(C2×C20), C60.261(C2×C4), C20.69(C2×C12), C22.7(C2×C60), C6.7(C5×M4(2)), C2.1(C15×M4(2)), (C2×C60).587C22, C10.12(C3×M4(2)), (C2×C6).37(C2×C20), (C2×C4).31(C2×C30), (C2×C20).133(C2×C6), (C2×C30).205(C2×C4), (C2×C10).57(C2×C12), (C2×C12).134(C2×C10), SmallGroup(480,200)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×C8⋊C4
G = < a,b,c | a15=b8=c4=1, ab=ba, ac=ca, cbc-1=b5 >
Subgroups: 88 in 80 conjugacy classes, 72 normal (24 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, C24, C2×C12, C2×C12, C30, C30, C8⋊C4, C40, C2×C20, C2×C20, C4×C12, C2×C24, C60, C60, C2×C30, C4×C20, C2×C40, C3×C8⋊C4, C120, C2×C60, C2×C60, C5×C8⋊C4, C4×C60, C2×C120, C15×C8⋊C4
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, C10, C12, C2×C6, C15, C42, M4(2), C20, C2×C10, C2×C12, C30, C8⋊C4, C2×C20, C4×C12, C3×M4(2), C60, C2×C30, C4×C20, C5×M4(2), C3×C8⋊C4, C2×C60, C5×C8⋊C4, C4×C60, C15×M4(2), C15×C8⋊C4
►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)
(1 135 20 435 187 232 420 98)(2 121 21 421 188 233 406 99)(3 122 22 422 189 234 407 100)(4 123 23 423 190 235 408 101)(5 124 24 424 191 236 409 102)(6 125 25 425 192 237 410 103)(7 126 26 426 193 238 411 104)(8 127 27 427 194 239 412 105)(9 128 28 428 195 240 413 91)(10 129 29 429 181 226 414 92)(11 130 30 430 182 227 415 93)(12 131 16 431 183 228 416 94)(13 132 17 432 184 229 417 95)(14 133 18 433 185 230 418 96)(15 134 19 434 186 231 419 97)(31 353 336 221 296 49 207 80)(32 354 337 222 297 50 208 81)(33 355 338 223 298 51 209 82)(34 356 339 224 299 52 210 83)(35 357 340 225 300 53 196 84)(36 358 341 211 286 54 197 85)(37 359 342 212 287 55 198 86)(38 360 343 213 288 56 199 87)(39 346 344 214 289 57 200 88)(40 347 345 215 290 58 201 89)(41 348 331 216 291 59 202 90)(42 349 332 217 292 60 203 76)(43 350 333 218 293 46 204 77)(44 351 334 219 294 47 205 78)(45 352 335 220 295 48 206 79)(61 398 176 459 153 442 284 384)(62 399 177 460 154 443 285 385)(63 400 178 461 155 444 271 386)(64 401 179 462 156 445 272 387)(65 402 180 463 157 446 273 388)(66 403 166 464 158 447 274 389)(67 404 167 465 159 448 275 390)(68 405 168 451 160 449 276 376)(69 391 169 452 161 450 277 377)(70 392 170 453 162 436 278 378)(71 393 171 454 163 437 279 379)(72 394 172 455 164 438 280 380)(73 395 173 456 165 439 281 381)(74 396 174 457 151 440 282 382)(75 397 175 458 152 441 283 383)(106 143 260 325 252 311 476 365)(107 144 261 326 253 312 477 366)(108 145 262 327 254 313 478 367)(109 146 263 328 255 314 479 368)(110 147 264 329 241 315 480 369)(111 148 265 330 242 301 466 370)(112 149 266 316 243 302 467 371)(113 150 267 317 244 303 468 372)(114 136 268 318 245 304 469 373)(115 137 269 319 246 305 470 374)(116 138 270 320 247 306 471 375)(117 139 256 321 248 307 472 361)(118 140 257 322 249 308 473 362)(119 141 258 323 250 309 474 363)(120 142 259 324 251 310 475 364)
(1 69 335 477)(2 70 336 478)(3 71 337 479)(4 72 338 480)(5 73 339 466)(6 74 340 467)(7 75 341 468)(8 61 342 469)(9 62 343 470)(10 63 344 471)(11 64 345 472)(12 65 331 473)(13 66 332 474)(14 67 333 475)(15 68 334 476)(16 180 291 118)(17 166 292 119)(18 167 293 120)(19 168 294 106)(20 169 295 107)(21 170 296 108)(22 171 297 109)(23 172 298 110)(24 173 299 111)(25 174 300 112)(26 175 286 113)(27 176 287 114)(28 177 288 115)(29 178 289 116)(30 179 290 117)(31 254 406 278)(32 255 407 279)(33 241 408 280)(34 242 409 281)(35 243 410 282)(36 244 411 283)(37 245 412 284)(38 246 413 285)(39 247 414 271)(40 248 415 272)(41 249 416 273)(42 250 417 274)(43 251 418 275)(44 252 419 276)(45 253 420 277)(46 310 433 390)(47 311 434 376)(48 312 435 377)(49 313 421 378)(50 314 422 379)(51 315 423 380)(52 301 424 381)(53 302 425 382)(54 303 426 383)(55 304 427 384)(56 305 428 385)(57 306 429 386)(58 307 430 387)(59 308 431 388)(60 309 432 389)(76 363 229 403)(77 364 230 404)(78 365 231 405)(79 366 232 391)(80 367 233 392)(81 368 234 393)(82 369 235 394)(83 370 236 395)(84 371 237 396)(85 372 238 397)(86 373 239 398)(87 374 240 399)(88 375 226 400)(89 361 227 401)(90 362 228 402)(91 460 360 137)(92 461 346 138)(93 462 347 139)(94 463 348 140)(95 464 349 141)(96 465 350 142)(97 451 351 143)(98 452 352 144)(99 453 353 145)(100 454 354 146)(101 455 355 147)(102 456 356 148)(103 457 357 149)(104 458 358 150)(105 459 359 136)(121 436 221 327)(122 437 222 328)(123 438 223 329)(124 439 224 330)(125 440 225 316)(126 441 211 317)(127 442 212 318)(128 443 213 319)(129 444 214 320)(130 445 215 321)(131 446 216 322)(132 447 217 323)(133 448 218 324)(134 449 219 325)(135 450 220 326)(151 196 266 192)(152 197 267 193)(153 198 268 194)(154 199 269 195)(155 200 270 181)(156 201 256 182)(157 202 257 183)(158 203 258 184)(159 204 259 185)(160 205 260 186)(161 206 261 187)(162 207 262 188)(163 208 263 189)(164 209 264 190)(165 210 265 191)
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), (1,135,20,435,187,232,420,98)(2,121,21,421,188,233,406,99)(3,122,22,422,189,234,407,100)(4,123,23,423,190,235,408,101)(5,124,24,424,191,236,409,102)(6,125,25,425,192,237,410,103)(7,126,26,426,193,238,411,104)(8,127,27,427,194,239,412,105)(9,128,28,428,195,240,413,91)(10,129,29,429,181,226,414,92)(11,130,30,430,182,227,415,93)(12,131,16,431,183,228,416,94)(13,132,17,432,184,229,417,95)(14,133,18,433,185,230,418,96)(15,134,19,434,186,231,419,97)(31,353,336,221,296,49,207,80)(32,354,337,222,297,50,208,81)(33,355,338,223,298,51,209,82)(34,356,339,224,299,52,210,83)(35,357,340,225,300,53,196,84)(36,358,341,211,286,54,197,85)(37,359,342,212,287,55,198,86)(38,360,343,213,288,56,199,87)(39,346,344,214,289,57,200,88)(40,347,345,215,290,58,201,89)(41,348,331,216,291,59,202,90)(42,349,332,217,292,60,203,76)(43,350,333,218,293,46,204,77)(44,351,334,219,294,47,205,78)(45,352,335,220,295,48,206,79)(61,398,176,459,153,442,284,384)(62,399,177,460,154,443,285,385)(63,400,178,461,155,444,271,386)(64,401,179,462,156,445,272,387)(65,402,180,463,157,446,273,388)(66,403,166,464,158,447,274,389)(67,404,167,465,159,448,275,390)(68,405,168,451,160,449,276,376)(69,391,169,452,161,450,277,377)(70,392,170,453,162,436,278,378)(71,393,171,454,163,437,279,379)(72,394,172,455,164,438,280,380)(73,395,173,456,165,439,281,381)(74,396,174,457,151,440,282,382)(75,397,175,458,152,441,283,383)(106,143,260,325,252,311,476,365)(107,144,261,326,253,312,477,366)(108,145,262,327,254,313,478,367)(109,146,263,328,255,314,479,368)(110,147,264,329,241,315,480,369)(111,148,265,330,242,301,466,370)(112,149,266,316,243,302,467,371)(113,150,267,317,244,303,468,372)(114,136,268,318,245,304,469,373)(115,137,269,319,246,305,470,374)(116,138,270,320,247,306,471,375)(117,139,256,321,248,307,472,361)(118,140,257,322,249,308,473,362)(119,141,258,323,250,309,474,363)(120,142,259,324,251,310,475,364), (1,69,335,477)(2,70,336,478)(3,71,337,479)(4,72,338,480)(5,73,339,466)(6,74,340,467)(7,75,341,468)(8,61,342,469)(9,62,343,470)(10,63,344,471)(11,64,345,472)(12,65,331,473)(13,66,332,474)(14,67,333,475)(15,68,334,476)(16,180,291,118)(17,166,292,119)(18,167,293,120)(19,168,294,106)(20,169,295,107)(21,170,296,108)(22,171,297,109)(23,172,298,110)(24,173,299,111)(25,174,300,112)(26,175,286,113)(27,176,287,114)(28,177,288,115)(29,178,289,116)(30,179,290,117)(31,254,406,278)(32,255,407,279)(33,241,408,280)(34,242,409,281)(35,243,410,282)(36,244,411,283)(37,245,412,284)(38,246,413,285)(39,247,414,271)(40,248,415,272)(41,249,416,273)(42,250,417,274)(43,251,418,275)(44,252,419,276)(45,253,420,277)(46,310,433,390)(47,311,434,376)(48,312,435,377)(49,313,421,378)(50,314,422,379)(51,315,423,380)(52,301,424,381)(53,302,425,382)(54,303,426,383)(55,304,427,384)(56,305,428,385)(57,306,429,386)(58,307,430,387)(59,308,431,388)(60,309,432,389)(76,363,229,403)(77,364,230,404)(78,365,231,405)(79,366,232,391)(80,367,233,392)(81,368,234,393)(82,369,235,394)(83,370,236,395)(84,371,237,396)(85,372,238,397)(86,373,239,398)(87,374,240,399)(88,375,226,400)(89,361,227,401)(90,362,228,402)(91,460,360,137)(92,461,346,138)(93,462,347,139)(94,463,348,140)(95,464,349,141)(96,465,350,142)(97,451,351,143)(98,452,352,144)(99,453,353,145)(100,454,354,146)(101,455,355,147)(102,456,356,148)(103,457,357,149)(104,458,358,150)(105,459,359,136)(121,436,221,327)(122,437,222,328)(123,438,223,329)(124,439,224,330)(125,440,225,316)(126,441,211,317)(127,442,212,318)(128,443,213,319)(129,444,214,320)(130,445,215,321)(131,446,216,322)(132,447,217,323)(133,448,218,324)(134,449,219,325)(135,450,220,326)(151,196,266,192)(152,197,267,193)(153,198,268,194)(154,199,269,195)(155,200,270,181)(156,201,256,182)(157,202,257,183)(158,203,258,184)(159,204,259,185)(160,205,260,186)(161,206,261,187)(162,207,262,188)(163,208,263,189)(164,209,264,190)(165,210,265,191)>;
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), (1,135,20,435,187,232,420,98)(2,121,21,421,188,233,406,99)(3,122,22,422,189,234,407,100)(4,123,23,423,190,235,408,101)(5,124,24,424,191,236,409,102)(6,125,25,425,192,237,410,103)(7,126,26,426,193,238,411,104)(8,127,27,427,194,239,412,105)(9,128,28,428,195,240,413,91)(10,129,29,429,181,226,414,92)(11,130,30,430,182,227,415,93)(12,131,16,431,183,228,416,94)(13,132,17,432,184,229,417,95)(14,133,18,433,185,230,418,96)(15,134,19,434,186,231,419,97)(31,353,336,221,296,49,207,80)(32,354,337,222,297,50,208,81)(33,355,338,223,298,51,209,82)(34,356,339,224,299,52,210,83)(35,357,340,225,300,53,196,84)(36,358,341,211,286,54,197,85)(37,359,342,212,287,55,198,86)(38,360,343,213,288,56,199,87)(39,346,344,214,289,57,200,88)(40,347,345,215,290,58,201,89)(41,348,331,216,291,59,202,90)(42,349,332,217,292,60,203,76)(43,350,333,218,293,46,204,77)(44,351,334,219,294,47,205,78)(45,352,335,220,295,48,206,79)(61,398,176,459,153,442,284,384)(62,399,177,460,154,443,285,385)(63,400,178,461,155,444,271,386)(64,401,179,462,156,445,272,387)(65,402,180,463,157,446,273,388)(66,403,166,464,158,447,274,389)(67,404,167,465,159,448,275,390)(68,405,168,451,160,449,276,376)(69,391,169,452,161,450,277,377)(70,392,170,453,162,436,278,378)(71,393,171,454,163,437,279,379)(72,394,172,455,164,438,280,380)(73,395,173,456,165,439,281,381)(74,396,174,457,151,440,282,382)(75,397,175,458,152,441,283,383)(106,143,260,325,252,311,476,365)(107,144,261,326,253,312,477,366)(108,145,262,327,254,313,478,367)(109,146,263,328,255,314,479,368)(110,147,264,329,241,315,480,369)(111,148,265,330,242,301,466,370)(112,149,266,316,243,302,467,371)(113,150,267,317,244,303,468,372)(114,136,268,318,245,304,469,373)(115,137,269,319,246,305,470,374)(116,138,270,320,247,306,471,375)(117,139,256,321,248,307,472,361)(118,140,257,322,249,308,473,362)(119,141,258,323,250,309,474,363)(120,142,259,324,251,310,475,364), (1,69,335,477)(2,70,336,478)(3,71,337,479)(4,72,338,480)(5,73,339,466)(6,74,340,467)(7,75,341,468)(8,61,342,469)(9,62,343,470)(10,63,344,471)(11,64,345,472)(12,65,331,473)(13,66,332,474)(14,67,333,475)(15,68,334,476)(16,180,291,118)(17,166,292,119)(18,167,293,120)(19,168,294,106)(20,169,295,107)(21,170,296,108)(22,171,297,109)(23,172,298,110)(24,173,299,111)(25,174,300,112)(26,175,286,113)(27,176,287,114)(28,177,288,115)(29,178,289,116)(30,179,290,117)(31,254,406,278)(32,255,407,279)(33,241,408,280)(34,242,409,281)(35,243,410,282)(36,244,411,283)(37,245,412,284)(38,246,413,285)(39,247,414,271)(40,248,415,272)(41,249,416,273)(42,250,417,274)(43,251,418,275)(44,252,419,276)(45,253,420,277)(46,310,433,390)(47,311,434,376)(48,312,435,377)(49,313,421,378)(50,314,422,379)(51,315,423,380)(52,301,424,381)(53,302,425,382)(54,303,426,383)(55,304,427,384)(56,305,428,385)(57,306,429,386)(58,307,430,387)(59,308,431,388)(60,309,432,389)(76,363,229,403)(77,364,230,404)(78,365,231,405)(79,366,232,391)(80,367,233,392)(81,368,234,393)(82,369,235,394)(83,370,236,395)(84,371,237,396)(85,372,238,397)(86,373,239,398)(87,374,240,399)(88,375,226,400)(89,361,227,401)(90,362,228,402)(91,460,360,137)(92,461,346,138)(93,462,347,139)(94,463,348,140)(95,464,349,141)(96,465,350,142)(97,451,351,143)(98,452,352,144)(99,453,353,145)(100,454,354,146)(101,455,355,147)(102,456,356,148)(103,457,357,149)(104,458,358,150)(105,459,359,136)(121,436,221,327)(122,437,222,328)(123,438,223,329)(124,439,224,330)(125,440,225,316)(126,441,211,317)(127,442,212,318)(128,443,213,319)(129,444,214,320)(130,445,215,321)(131,446,216,322)(132,447,217,323)(133,448,218,324)(134,449,219,325)(135,450,220,326)(151,196,266,192)(152,197,267,193)(153,198,268,194)(154,199,269,195)(155,200,270,181)(156,201,256,182)(157,202,257,183)(158,203,258,184)(159,204,259,185)(160,205,260,186)(161,206,261,187)(162,207,262,188)(163,208,263,189)(164,209,264,190)(165,210,265,191) );
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)], [(1,135,20,435,187,232,420,98),(2,121,21,421,188,233,406,99),(3,122,22,422,189,234,407,100),(4,123,23,423,190,235,408,101),(5,124,24,424,191,236,409,102),(6,125,25,425,192,237,410,103),(7,126,26,426,193,238,411,104),(8,127,27,427,194,239,412,105),(9,128,28,428,195,240,413,91),(10,129,29,429,181,226,414,92),(11,130,30,430,182,227,415,93),(12,131,16,431,183,228,416,94),(13,132,17,432,184,229,417,95),(14,133,18,433,185,230,418,96),(15,134,19,434,186,231,419,97),(31,353,336,221,296,49,207,80),(32,354,337,222,297,50,208,81),(33,355,338,223,298,51,209,82),(34,356,339,224,299,52,210,83),(35,357,340,225,300,53,196,84),(36,358,341,211,286,54,197,85),(37,359,342,212,287,55,198,86),(38,360,343,213,288,56,199,87),(39,346,344,214,289,57,200,88),(40,347,345,215,290,58,201,89),(41,348,331,216,291,59,202,90),(42,349,332,217,292,60,203,76),(43,350,333,218,293,46,204,77),(44,351,334,219,294,47,205,78),(45,352,335,220,295,48,206,79),(61,398,176,459,153,442,284,384),(62,399,177,460,154,443,285,385),(63,400,178,461,155,444,271,386),(64,401,179,462,156,445,272,387),(65,402,180,463,157,446,273,388),(66,403,166,464,158,447,274,389),(67,404,167,465,159,448,275,390),(68,405,168,451,160,449,276,376),(69,391,169,452,161,450,277,377),(70,392,170,453,162,436,278,378),(71,393,171,454,163,437,279,379),(72,394,172,455,164,438,280,380),(73,395,173,456,165,439,281,381),(74,396,174,457,151,440,282,382),(75,397,175,458,152,441,283,383),(106,143,260,325,252,311,476,365),(107,144,261,326,253,312,477,366),(108,145,262,327,254,313,478,367),(109,146,263,328,255,314,479,368),(110,147,264,329,241,315,480,369),(111,148,265,330,242,301,466,370),(112,149,266,316,243,302,467,371),(113,150,267,317,244,303,468,372),(114,136,268,318,245,304,469,373),(115,137,269,319,246,305,470,374),(116,138,270,320,247,306,471,375),(117,139,256,321,248,307,472,361),(118,140,257,322,249,308,473,362),(119,141,258,323,250,309,474,363),(120,142,259,324,251,310,475,364)], [(1,69,335,477),(2,70,336,478),(3,71,337,479),(4,72,338,480),(5,73,339,466),(6,74,340,467),(7,75,341,468),(8,61,342,469),(9,62,343,470),(10,63,344,471),(11,64,345,472),(12,65,331,473),(13,66,332,474),(14,67,333,475),(15,68,334,476),(16,180,291,118),(17,166,292,119),(18,167,293,120),(19,168,294,106),(20,169,295,107),(21,170,296,108),(22,171,297,109),(23,172,298,110),(24,173,299,111),(25,174,300,112),(26,175,286,113),(27,176,287,114),(28,177,288,115),(29,178,289,116),(30,179,290,117),(31,254,406,278),(32,255,407,279),(33,241,408,280),(34,242,409,281),(35,243,410,282),(36,244,411,283),(37,245,412,284),(38,246,413,285),(39,247,414,271),(40,248,415,272),(41,249,416,273),(42,250,417,274),(43,251,418,275),(44,252,419,276),(45,253,420,277),(46,310,433,390),(47,311,434,376),(48,312,435,377),(49,313,421,378),(50,314,422,379),(51,315,423,380),(52,301,424,381),(53,302,425,382),(54,303,426,383),(55,304,427,384),(56,305,428,385),(57,306,429,386),(58,307,430,387),(59,308,431,388),(60,309,432,389),(76,363,229,403),(77,364,230,404),(78,365,231,405),(79,366,232,391),(80,367,233,392),(81,368,234,393),(82,369,235,394),(83,370,236,395),(84,371,237,396),(85,372,238,397),(86,373,239,398),(87,374,240,399),(88,375,226,400),(89,361,227,401),(90,362,228,402),(91,460,360,137),(92,461,346,138),(93,462,347,139),(94,463,348,140),(95,464,349,141),(96,465,350,142),(97,451,351,143),(98,452,352,144),(99,453,353,145),(100,454,354,146),(101,455,355,147),(102,456,356,148),(103,457,357,149),(104,458,358,150),(105,459,359,136),(121,436,221,327),(122,437,222,328),(123,438,223,329),(124,439,224,330),(125,440,225,316),(126,441,211,317),(127,442,212,318),(128,443,213,319),(129,444,214,320),(130,445,215,321),(131,446,216,322),(132,447,217,323),(133,448,218,324),(134,449,219,325),(135,450,220,326),(151,196,266,192),(152,197,267,193),(153,198,268,194),(154,199,269,195),(155,200,270,181),(156,201,256,182),(157,202,257,183),(158,203,258,184),(159,204,259,185),(160,205,260,186),(161,206,261,187),(162,207,262,188),(163,208,263,189),(164,209,264,190),(165,210,265,191)]])
300 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 8A | ··· | 8H | 10A | ··· | 10L | 12A | ··· | 12H | 12I | ··· | 12P | 15A | ··· | 15H | 20A | ··· | 20P | 20Q | ··· | 20AF | 24A | ··· | 24P | 30A | ··· | 30X | 40A | ··· | 40AF | 60A | ··· | 60AF | 60AG | ··· | 60BL | 120A | ··· | 120BL |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 8 | ··· | 8 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
300 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C5 | C6 | C6 | C10 | C10 | C12 | C12 | C15 | C20 | C20 | C30 | C30 | C60 | C60 | M4(2) | C3×M4(2) | C5×M4(2) | C15×M4(2) |
| kernel | C15×C8⋊C4 | C4×C60 | C2×C120 | C5×C8⋊C4 | C120 | C2×C60 | C3×C8⋊C4 | C4×C20 | C2×C40 | C4×C12 | C2×C24 | C40 | C2×C20 | C8⋊C4 | C24 | C2×C12 | C42 | C2×C8 | C8 | C2×C4 | C30 | C10 | C6 | C2 |
| # reps | 1 | 1 | 2 | 2 | 8 | 4 | 4 | 2 | 4 | 4 | 8 | 16 | 8 | 8 | 32 | 16 | 8 | 16 | 64 | 32 | 4 | 8 | 16 | 32 |
Matrix representation of C15×C8⋊C4 ►in GL3(𝔽241) generated by
| 15 | 0 | 0 |
| 0 | 87 | 0 |
| 0 | 0 | 87 |
| 177 | 0 | 0 |
| 0 | 188 | 121 |
| 0 | 53 | 53 |
| 177 | 0 | 0 |
| 0 | 240 | 239 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(241))| [15,0,0,0,87,0,0,0,87],[177,0,0,0,188,53,0,121,53],[177,0,0,0,240,0,0,239,1] >;
C15×C8⋊C4 in GAP, Magma, Sage, TeX
C_{15}\times C_8\rtimes C_4 % in TeX
G:=Group("C15xC8:C4"); // GroupNames label
G:=SmallGroup(480,200);
// by ID
G=gap.SmallGroup(480,200);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,420,3389,848,172]);
// Polycyclic
G:=Group<a,b,c|a^15=b^8=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations