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G = C15×C8⋊C4order 480 = 25·3·5

Direct product of C15 and C8⋊C4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: C15×C8⋊C4, C83C60, C247C20, C12023C4, C4011C12, 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

C1C2 — C15×C8⋊C4
C1C2C22C2×C4C2×C20C2×C60C2×C120 — C15×C8⋊C4
C1C2 — C15×C8⋊C4
C1C2×C60 — C15×C8⋊C4

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 [×2], C3, C4 [×2], C4 [×2], C22, C5, C6, C6 [×2], C8 [×4], C2×C4, C2×C4 [×2], C10, C10 [×2], C12 [×2], C12 [×2], C2×C6, C15, C42, C2×C8 [×2], C20 [×2], C20 [×2], C2×C10, C24 [×4], C2×C12, C2×C12 [×2], C30, C30 [×2], C8⋊C4, C40 [×4], C2×C20, C2×C20 [×2], C4×C12, C2×C24 [×2], C60 [×2], C60 [×2], C2×C30, C4×C20, C2×C40 [×2], C3×C8⋊C4, C120 [×4], C2×C60, C2×C60 [×2], C5×C8⋊C4, C4×C60, C2×C120 [×2], C15×C8⋊C4
Quotients: C1, C2 [×3], C3, C4 [×6], C22, C5, C6 [×3], C2×C4 [×3], C10 [×3], C12 [×6], C2×C6, C15, C42, M4(2) [×2], C20 [×6], C2×C10, C2×C12 [×3], C30 [×3], C8⋊C4, C2×C20 [×3], C4×C12, C3×M4(2) [×2], C60 [×6], C2×C30, C4×C20, C5×M4(2) [×2], C3×C8⋊C4, C2×C60 [×3], C5×C8⋊C4, C4×C60, C15×M4(2) [×2], C15×C8⋊C4

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

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

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

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

300 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F4G4H5A5B5C5D6A···6F8A···8H10A···10L12A···12H12I···12P15A···15H20A···20P20Q···20AF24A···24P30A···30X40A···40AF60A···60AF60AG···60BL120A···120BL
order1222334444444455556···68···810···1012···1212···1215···1520···2020···2024···2430···3040···4060···6060···60120···120
size1111111111222211111···12···21···11···12···21···11···12···22···21···12···21···12···22···2

300 irreducible representations

dim111111111111111111112222
type+++
imageC1C2C2C3C4C4C5C6C6C10C10C12C12C15C20C20C30C30C60C60M4(2)C3×M4(2)C5×M4(2)C15×M4(2)
kernelC15×C8⋊C4C4×C60C2×C120C5×C8⋊C4C120C2×C60C3×C8⋊C4C4×C20C2×C40C4×C12C2×C24C40C2×C20C8⋊C4C24C2×C12C42C2×C8C8C2×C4C30C10C6C2
# reps11228442448168832168166432481632

Matrix representation of C15×C8⋊C4 in GL3(𝔽241) generated by

1500
0870
0087
,
17700
0188121
05353
,
17700
0240239
001
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

׿
×
𝔽