direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C15×C4⋊C8, C4⋊C120, C12⋊3C40, C60⋊11C8, C20⋊5C24, C60.40Q8, C60.252D4, C42.2C30, C30.45M4(2), (C2×C4).4C60, (C2×C8).2C30, (C2×C40).4C6, (C4×C20).8C6, C4.4(Q8×C15), C6.12(C2×C40), (C4×C60).20C2, (C2×C60).53C4, (C4×C12).8C10, (C2×C120).8C2, (C2×C24).4C10, C30.74(C2×C8), (C2×C12).8C20, C2.2(C2×C120), C4.18(D4×C15), C12.67(C5×D4), C20.67(C3×D4), C30.63(C4⋊C4), C20.12(C3×Q8), C12.12(C5×Q8), C10.21(C2×C24), (C2×C20).19C12, C22.9(C2×C60), C6.9(C5×M4(2)), C2.3(C15×M4(2)), (C2×C60).589C22, C10.14(C3×M4(2)), C6.11(C5×C4⋊C4), C2.2(C15×C4⋊C4), C10.18(C3×C4⋊C4), (C2×C6).39(C2×C20), (C2×C4).33(C2×C30), (C2×C30).207(C2×C4), (C2×C10).59(C2×C12), (C2×C20).135(C2×C6), (C2×C12).136(C2×C10), SmallGroup(480,208)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×C4⋊C8
G = < a,b,c | a15=b4=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 88 in 76 conjugacy classes, 64 normal (48 characteristic)
C1, C2, C3, C4, C4, C4, C22, C5, C6, C8, C2×C4, C10, C12, C12, C12, C2×C6, C15, C42, C2×C8, C20, C20, C20, C2×C10, C24, C2×C12, C30, C4⋊C8, C40, C2×C20, C4×C12, C2×C24, C60, C60, C60, C2×C30, C4×C20, C2×C40, C3×C4⋊C8, C120, C2×C60, C5×C4⋊C8, C4×C60, C2×C120, C15×C4⋊C8
Quotients: C1, C2, C3, C4, C22, C5, C6, C8, C2×C4, D4, Q8, C10, C12, C2×C6, C15, C4⋊C4, C2×C8, M4(2), C20, C2×C10, C24, C2×C12, C3×D4, C3×Q8, C30, C4⋊C8, C40, C2×C20, C5×D4, C5×Q8, C3×C4⋊C4, C2×C24, C3×M4(2), C60, C2×C30, C5×C4⋊C4, C2×C40, C5×M4(2), C3×C4⋊C8, C120, C2×C60, D4×C15, Q8×C15, C5×C4⋊C8, C15×C4⋊C4, C2×C120, C15×M4(2), C15×C4⋊C8
►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)
(1 171 255 312)(2 172 241 313)(3 173 242 314)(4 174 243 315)(5 175 244 301)(6 176 245 302)(7 177 246 303)(8 178 247 304)(9 179 248 305)(10 180 249 306)(11 166 250 307)(12 167 251 308)(13 168 252 309)(14 169 253 310)(15 170 254 311)(16 475 340 407)(17 476 341 408)(18 477 342 409)(19 478 343 410)(20 479 344 411)(21 480 345 412)(22 466 331 413)(23 467 332 414)(24 468 333 415)(25 469 334 416)(26 470 335 417)(27 471 336 418)(28 472 337 419)(29 473 338 420)(30 474 339 406)(31 207 65 50)(32 208 66 51)(33 209 67 52)(34 210 68 53)(35 196 69 54)(36 197 70 55)(37 198 71 56)(38 199 72 57)(39 200 73 58)(40 201 74 59)(41 202 75 60)(42 203 61 46)(43 204 62 47)(44 205 63 48)(45 206 64 49)(76 140 280 95)(77 141 281 96)(78 142 282 97)(79 143 283 98)(80 144 284 99)(81 145 285 100)(82 146 271 101)(83 147 272 102)(84 148 273 103)(85 149 274 104)(86 150 275 105)(87 136 276 91)(88 137 277 92)(89 138 278 93)(90 139 279 94)(106 211 156 186)(107 212 157 187)(108 213 158 188)(109 214 159 189)(110 215 160 190)(111 216 161 191)(112 217 162 192)(113 218 163 193)(114 219 164 194)(115 220 165 195)(116 221 151 181)(117 222 152 182)(118 223 153 183)(119 224 154 184)(120 225 155 185)(121 291 235 461)(122 292 236 462)(123 293 237 463)(124 294 238 464)(125 295 239 465)(126 296 240 451)(127 297 226 452)(128 298 227 453)(129 299 228 454)(130 300 229 455)(131 286 230 456)(132 287 231 457)(133 288 232 458)(134 289 233 459)(135 290 234 460)(256 320 449 396)(257 321 450 397)(258 322 436 398)(259 323 437 399)(260 324 438 400)(261 325 439 401)(262 326 440 402)(263 327 441 403)(264 328 442 404)(265 329 443 405)(266 330 444 391)(267 316 445 392)(268 317 446 393)(269 318 447 394)(270 319 448 395)(346 426 369 389)(347 427 370 390)(348 428 371 376)(349 429 372 377)(350 430 373 378)(351 431 374 379)(352 432 375 380)(353 433 361 381)(354 434 362 382)(355 435 363 383)(356 421 364 384)(357 422 365 385)(358 423 366 386)(359 424 367 387)(360 425 368 388)
(1 350 49 214 263 235 88 16)(2 351 50 215 264 236 89 17)(3 352 51 216 265 237 90 18)(4 353 52 217 266 238 76 19)(5 354 53 218 267 239 77 20)(6 355 54 219 268 240 78 21)(7 356 55 220 269 226 79 22)(8 357 56 221 270 227 80 23)(9 358 57 222 256 228 81 24)(10 359 58 223 257 229 82 25)(11 360 59 224 258 230 83 26)(12 346 60 225 259 231 84 27)(13 347 46 211 260 232 85 28)(14 348 47 212 261 233 86 29)(15 349 48 213 262 234 87 30)(31 110 328 292 138 408 172 379)(32 111 329 293 139 409 173 380)(33 112 330 294 140 410 174 381)(34 113 316 295 141 411 175 382)(35 114 317 296 142 412 176 383)(36 115 318 297 143 413 177 384)(37 116 319 298 144 414 178 385)(38 117 320 299 145 415 179 386)(39 118 321 300 146 416 180 387)(40 119 322 286 147 417 166 388)(41 120 323 287 148 418 167 389)(42 106 324 288 149 419 168 390)(43 107 325 289 150 420 169 376)(44 108 326 290 136 406 170 377)(45 109 327 291 137 407 171 378)(61 156 400 458 104 472 309 427)(62 157 401 459 105 473 310 428)(63 158 402 460 91 474 311 429)(64 159 403 461 92 475 312 430)(65 160 404 462 93 476 313 431)(66 161 405 463 94 477 314 432)(67 162 391 464 95 478 315 433)(68 163 392 465 96 479 301 434)(69 164 393 451 97 480 302 435)(70 165 394 452 98 466 303 421)(71 151 395 453 99 467 304 422)(72 152 396 454 100 468 305 423)(73 153 397 455 101 469 306 424)(74 154 398 456 102 470 307 425)(75 155 399 457 103 471 308 426)(121 277 340 255 373 206 189 441)(122 278 341 241 374 207 190 442)(123 279 342 242 375 208 191 443)(124 280 343 243 361 209 192 444)(125 281 344 244 362 210 193 445)(126 282 345 245 363 196 194 446)(127 283 331 246 364 197 195 447)(128 284 332 247 365 198 181 448)(129 285 333 248 366 199 182 449)(130 271 334 249 367 200 183 450)(131 272 335 250 368 201 184 436)(132 273 336 251 369 202 185 437)(133 274 337 252 370 203 186 438)(134 275 338 253 371 204 187 439)(135 276 339 254 372 205 188 440)
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), (1,171,255,312)(2,172,241,313)(3,173,242,314)(4,174,243,315)(5,175,244,301)(6,176,245,302)(7,177,246,303)(8,178,247,304)(9,179,248,305)(10,180,249,306)(11,166,250,307)(12,167,251,308)(13,168,252,309)(14,169,253,310)(15,170,254,311)(16,475,340,407)(17,476,341,408)(18,477,342,409)(19,478,343,410)(20,479,344,411)(21,480,345,412)(22,466,331,413)(23,467,332,414)(24,468,333,415)(25,469,334,416)(26,470,335,417)(27,471,336,418)(28,472,337,419)(29,473,338,420)(30,474,339,406)(31,207,65,50)(32,208,66,51)(33,209,67,52)(34,210,68,53)(35,196,69,54)(36,197,70,55)(37,198,71,56)(38,199,72,57)(39,200,73,58)(40,201,74,59)(41,202,75,60)(42,203,61,46)(43,204,62,47)(44,205,63,48)(45,206,64,49)(76,140,280,95)(77,141,281,96)(78,142,282,97)(79,143,283,98)(80,144,284,99)(81,145,285,100)(82,146,271,101)(83,147,272,102)(84,148,273,103)(85,149,274,104)(86,150,275,105)(87,136,276,91)(88,137,277,92)(89,138,278,93)(90,139,279,94)(106,211,156,186)(107,212,157,187)(108,213,158,188)(109,214,159,189)(110,215,160,190)(111,216,161,191)(112,217,162,192)(113,218,163,193)(114,219,164,194)(115,220,165,195)(116,221,151,181)(117,222,152,182)(118,223,153,183)(119,224,154,184)(120,225,155,185)(121,291,235,461)(122,292,236,462)(123,293,237,463)(124,294,238,464)(125,295,239,465)(126,296,240,451)(127,297,226,452)(128,298,227,453)(129,299,228,454)(130,300,229,455)(131,286,230,456)(132,287,231,457)(133,288,232,458)(134,289,233,459)(135,290,234,460)(256,320,449,396)(257,321,450,397)(258,322,436,398)(259,323,437,399)(260,324,438,400)(261,325,439,401)(262,326,440,402)(263,327,441,403)(264,328,442,404)(265,329,443,405)(266,330,444,391)(267,316,445,392)(268,317,446,393)(269,318,447,394)(270,319,448,395)(346,426,369,389)(347,427,370,390)(348,428,371,376)(349,429,372,377)(350,430,373,378)(351,431,374,379)(352,432,375,380)(353,433,361,381)(354,434,362,382)(355,435,363,383)(356,421,364,384)(357,422,365,385)(358,423,366,386)(359,424,367,387)(360,425,368,388), (1,350,49,214,263,235,88,16)(2,351,50,215,264,236,89,17)(3,352,51,216,265,237,90,18)(4,353,52,217,266,238,76,19)(5,354,53,218,267,239,77,20)(6,355,54,219,268,240,78,21)(7,356,55,220,269,226,79,22)(8,357,56,221,270,227,80,23)(9,358,57,222,256,228,81,24)(10,359,58,223,257,229,82,25)(11,360,59,224,258,230,83,26)(12,346,60,225,259,231,84,27)(13,347,46,211,260,232,85,28)(14,348,47,212,261,233,86,29)(15,349,48,213,262,234,87,30)(31,110,328,292,138,408,172,379)(32,111,329,293,139,409,173,380)(33,112,330,294,140,410,174,381)(34,113,316,295,141,411,175,382)(35,114,317,296,142,412,176,383)(36,115,318,297,143,413,177,384)(37,116,319,298,144,414,178,385)(38,117,320,299,145,415,179,386)(39,118,321,300,146,416,180,387)(40,119,322,286,147,417,166,388)(41,120,323,287,148,418,167,389)(42,106,324,288,149,419,168,390)(43,107,325,289,150,420,169,376)(44,108,326,290,136,406,170,377)(45,109,327,291,137,407,171,378)(61,156,400,458,104,472,309,427)(62,157,401,459,105,473,310,428)(63,158,402,460,91,474,311,429)(64,159,403,461,92,475,312,430)(65,160,404,462,93,476,313,431)(66,161,405,463,94,477,314,432)(67,162,391,464,95,478,315,433)(68,163,392,465,96,479,301,434)(69,164,393,451,97,480,302,435)(70,165,394,452,98,466,303,421)(71,151,395,453,99,467,304,422)(72,152,396,454,100,468,305,423)(73,153,397,455,101,469,306,424)(74,154,398,456,102,470,307,425)(75,155,399,457,103,471,308,426)(121,277,340,255,373,206,189,441)(122,278,341,241,374,207,190,442)(123,279,342,242,375,208,191,443)(124,280,343,243,361,209,192,444)(125,281,344,244,362,210,193,445)(126,282,345,245,363,196,194,446)(127,283,331,246,364,197,195,447)(128,284,332,247,365,198,181,448)(129,285,333,248,366,199,182,449)(130,271,334,249,367,200,183,450)(131,272,335,250,368,201,184,436)(132,273,336,251,369,202,185,437)(133,274,337,252,370,203,186,438)(134,275,338,253,371,204,187,439)(135,276,339,254,372,205,188,440)>;
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), (1,171,255,312)(2,172,241,313)(3,173,242,314)(4,174,243,315)(5,175,244,301)(6,176,245,302)(7,177,246,303)(8,178,247,304)(9,179,248,305)(10,180,249,306)(11,166,250,307)(12,167,251,308)(13,168,252,309)(14,169,253,310)(15,170,254,311)(16,475,340,407)(17,476,341,408)(18,477,342,409)(19,478,343,410)(20,479,344,411)(21,480,345,412)(22,466,331,413)(23,467,332,414)(24,468,333,415)(25,469,334,416)(26,470,335,417)(27,471,336,418)(28,472,337,419)(29,473,338,420)(30,474,339,406)(31,207,65,50)(32,208,66,51)(33,209,67,52)(34,210,68,53)(35,196,69,54)(36,197,70,55)(37,198,71,56)(38,199,72,57)(39,200,73,58)(40,201,74,59)(41,202,75,60)(42,203,61,46)(43,204,62,47)(44,205,63,48)(45,206,64,49)(76,140,280,95)(77,141,281,96)(78,142,282,97)(79,143,283,98)(80,144,284,99)(81,145,285,100)(82,146,271,101)(83,147,272,102)(84,148,273,103)(85,149,274,104)(86,150,275,105)(87,136,276,91)(88,137,277,92)(89,138,278,93)(90,139,279,94)(106,211,156,186)(107,212,157,187)(108,213,158,188)(109,214,159,189)(110,215,160,190)(111,216,161,191)(112,217,162,192)(113,218,163,193)(114,219,164,194)(115,220,165,195)(116,221,151,181)(117,222,152,182)(118,223,153,183)(119,224,154,184)(120,225,155,185)(121,291,235,461)(122,292,236,462)(123,293,237,463)(124,294,238,464)(125,295,239,465)(126,296,240,451)(127,297,226,452)(128,298,227,453)(129,299,228,454)(130,300,229,455)(131,286,230,456)(132,287,231,457)(133,288,232,458)(134,289,233,459)(135,290,234,460)(256,320,449,396)(257,321,450,397)(258,322,436,398)(259,323,437,399)(260,324,438,400)(261,325,439,401)(262,326,440,402)(263,327,441,403)(264,328,442,404)(265,329,443,405)(266,330,444,391)(267,316,445,392)(268,317,446,393)(269,318,447,394)(270,319,448,395)(346,426,369,389)(347,427,370,390)(348,428,371,376)(349,429,372,377)(350,430,373,378)(351,431,374,379)(352,432,375,380)(353,433,361,381)(354,434,362,382)(355,435,363,383)(356,421,364,384)(357,422,365,385)(358,423,366,386)(359,424,367,387)(360,425,368,388), (1,350,49,214,263,235,88,16)(2,351,50,215,264,236,89,17)(3,352,51,216,265,237,90,18)(4,353,52,217,266,238,76,19)(5,354,53,218,267,239,77,20)(6,355,54,219,268,240,78,21)(7,356,55,220,269,226,79,22)(8,357,56,221,270,227,80,23)(9,358,57,222,256,228,81,24)(10,359,58,223,257,229,82,25)(11,360,59,224,258,230,83,26)(12,346,60,225,259,231,84,27)(13,347,46,211,260,232,85,28)(14,348,47,212,261,233,86,29)(15,349,48,213,262,234,87,30)(31,110,328,292,138,408,172,379)(32,111,329,293,139,409,173,380)(33,112,330,294,140,410,174,381)(34,113,316,295,141,411,175,382)(35,114,317,296,142,412,176,383)(36,115,318,297,143,413,177,384)(37,116,319,298,144,414,178,385)(38,117,320,299,145,415,179,386)(39,118,321,300,146,416,180,387)(40,119,322,286,147,417,166,388)(41,120,323,287,148,418,167,389)(42,106,324,288,149,419,168,390)(43,107,325,289,150,420,169,376)(44,108,326,290,136,406,170,377)(45,109,327,291,137,407,171,378)(61,156,400,458,104,472,309,427)(62,157,401,459,105,473,310,428)(63,158,402,460,91,474,311,429)(64,159,403,461,92,475,312,430)(65,160,404,462,93,476,313,431)(66,161,405,463,94,477,314,432)(67,162,391,464,95,478,315,433)(68,163,392,465,96,479,301,434)(69,164,393,451,97,480,302,435)(70,165,394,452,98,466,303,421)(71,151,395,453,99,467,304,422)(72,152,396,454,100,468,305,423)(73,153,397,455,101,469,306,424)(74,154,398,456,102,470,307,425)(75,155,399,457,103,471,308,426)(121,277,340,255,373,206,189,441)(122,278,341,241,374,207,190,442)(123,279,342,242,375,208,191,443)(124,280,343,243,361,209,192,444)(125,281,344,244,362,210,193,445)(126,282,345,245,363,196,194,446)(127,283,331,246,364,197,195,447)(128,284,332,247,365,198,181,448)(129,285,333,248,366,199,182,449)(130,271,334,249,367,200,183,450)(131,272,335,250,368,201,184,436)(132,273,336,251,369,202,185,437)(133,274,337,252,370,203,186,438)(134,275,338,253,371,204,187,439)(135,276,339,254,372,205,188,440) );
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)], [(1,171,255,312),(2,172,241,313),(3,173,242,314),(4,174,243,315),(5,175,244,301),(6,176,245,302),(7,177,246,303),(8,178,247,304),(9,179,248,305),(10,180,249,306),(11,166,250,307),(12,167,251,308),(13,168,252,309),(14,169,253,310),(15,170,254,311),(16,475,340,407),(17,476,341,408),(18,477,342,409),(19,478,343,410),(20,479,344,411),(21,480,345,412),(22,466,331,413),(23,467,332,414),(24,468,333,415),(25,469,334,416),(26,470,335,417),(27,471,336,418),(28,472,337,419),(29,473,338,420),(30,474,339,406),(31,207,65,50),(32,208,66,51),(33,209,67,52),(34,210,68,53),(35,196,69,54),(36,197,70,55),(37,198,71,56),(38,199,72,57),(39,200,73,58),(40,201,74,59),(41,202,75,60),(42,203,61,46),(43,204,62,47),(44,205,63,48),(45,206,64,49),(76,140,280,95),(77,141,281,96),(78,142,282,97),(79,143,283,98),(80,144,284,99),(81,145,285,100),(82,146,271,101),(83,147,272,102),(84,148,273,103),(85,149,274,104),(86,150,275,105),(87,136,276,91),(88,137,277,92),(89,138,278,93),(90,139,279,94),(106,211,156,186),(107,212,157,187),(108,213,158,188),(109,214,159,189),(110,215,160,190),(111,216,161,191),(112,217,162,192),(113,218,163,193),(114,219,164,194),(115,220,165,195),(116,221,151,181),(117,222,152,182),(118,223,153,183),(119,224,154,184),(120,225,155,185),(121,291,235,461),(122,292,236,462),(123,293,237,463),(124,294,238,464),(125,295,239,465),(126,296,240,451),(127,297,226,452),(128,298,227,453),(129,299,228,454),(130,300,229,455),(131,286,230,456),(132,287,231,457),(133,288,232,458),(134,289,233,459),(135,290,234,460),(256,320,449,396),(257,321,450,397),(258,322,436,398),(259,323,437,399),(260,324,438,400),(261,325,439,401),(262,326,440,402),(263,327,441,403),(264,328,442,404),(265,329,443,405),(266,330,444,391),(267,316,445,392),(268,317,446,393),(269,318,447,394),(270,319,448,395),(346,426,369,389),(347,427,370,390),(348,428,371,376),(349,429,372,377),(350,430,373,378),(351,431,374,379),(352,432,375,380),(353,433,361,381),(354,434,362,382),(355,435,363,383),(356,421,364,384),(357,422,365,385),(358,423,366,386),(359,424,367,387),(360,425,368,388)], [(1,350,49,214,263,235,88,16),(2,351,50,215,264,236,89,17),(3,352,51,216,265,237,90,18),(4,353,52,217,266,238,76,19),(5,354,53,218,267,239,77,20),(6,355,54,219,268,240,78,21),(7,356,55,220,269,226,79,22),(8,357,56,221,270,227,80,23),(9,358,57,222,256,228,81,24),(10,359,58,223,257,229,82,25),(11,360,59,224,258,230,83,26),(12,346,60,225,259,231,84,27),(13,347,46,211,260,232,85,28),(14,348,47,212,261,233,86,29),(15,349,48,213,262,234,87,30),(31,110,328,292,138,408,172,379),(32,111,329,293,139,409,173,380),(33,112,330,294,140,410,174,381),(34,113,316,295,141,411,175,382),(35,114,317,296,142,412,176,383),(36,115,318,297,143,413,177,384),(37,116,319,298,144,414,178,385),(38,117,320,299,145,415,179,386),(39,118,321,300,146,416,180,387),(40,119,322,286,147,417,166,388),(41,120,323,287,148,418,167,389),(42,106,324,288,149,419,168,390),(43,107,325,289,150,420,169,376),(44,108,326,290,136,406,170,377),(45,109,327,291,137,407,171,378),(61,156,400,458,104,472,309,427),(62,157,401,459,105,473,310,428),(63,158,402,460,91,474,311,429),(64,159,403,461,92,475,312,430),(65,160,404,462,93,476,313,431),(66,161,405,463,94,477,314,432),(67,162,391,464,95,478,315,433),(68,163,392,465,96,479,301,434),(69,164,393,451,97,480,302,435),(70,165,394,452,98,466,303,421),(71,151,395,453,99,467,304,422),(72,152,396,454,100,468,305,423),(73,153,397,455,101,469,306,424),(74,154,398,456,102,470,307,425),(75,155,399,457,103,471,308,426),(121,277,340,255,373,206,189,441),(122,278,341,241,374,207,190,442),(123,279,342,242,375,208,191,443),(124,280,343,243,361,209,192,444),(125,281,344,244,362,210,193,445),(126,282,345,245,363,196,194,446),(127,283,331,246,364,197,195,447),(128,284,332,247,365,198,181,448),(129,285,333,248,366,199,182,449),(130,271,334,249,367,200,183,450),(131,272,335,250,368,201,184,436),(132,273,336,251,369,202,185,437),(133,274,337,252,370,203,186,438),(134,275,338,253,371,204,187,439),(135,276,339,254,372,205,188,440)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C5 | C6 | C6 | C8 | C10 | C10 | C12 | C15 | C20 | C24 | C30 | C30 | C40 | C60 | C120 | D4 | Q8 | M4(2) | C3×D4 | C3×Q8 | C5×D4 | C5×Q8 | C3×M4(2) | C5×M4(2) | D4×C15 | Q8×C15 | C15×M4(2) |
| kernel | C15×C4⋊C8 | C4×C60 | C2×C120 | C5×C4⋊C8 | C2×C60 | C3×C4⋊C8 | C4×C20 | C2×C40 | C60 | C4×C12 | C2×C24 | C2×C20 | C4⋊C8 | C2×C12 | C20 | C42 | C2×C8 | C12 | C2×C4 | C4 | C60 | C60 | C30 | C20 | C20 | C12 | C12 | C10 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 4 | 8 | 4 | 8 | 8 | 8 | 16 | 16 | 8 | 16 | 32 | 32 | 64 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 16 |
Matrix representation of C15×C4⋊C8 ►in GL4(𝔽241) generated by
| 225 | 0 | 0 | 0 |
| 0 | 91 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 240 | 2 |
| 0 | 0 | 240 | 1 |
| 233 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 |
| 0 | 0 | 95 | 43 |
| 0 | 0 | 237 | 146 |
G:=sub<GL(4,GF(241))| [225,0,0,0,0,91,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,240,0,0,0,0,240,240,0,0,2,1],[233,0,0,0,0,177,0,0,0,0,95,237,0,0,43,146] >;
C15×C4⋊C8 in GAP, Magma, Sage, TeX
C_{15}\times C_4\rtimes C_8 % in TeX
G:=Group("C15xC4:C8"); // GroupNames label
G:=SmallGroup(480,208);
// by ID
G=gap.SmallGroup(480,208);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,840,869,428,124]);
// Polycyclic
G:=Group<a,b,c|a^15=b^4=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations