Copied to
clipboard

G = C15×C4⋊C8order 480 = 25·3·5

Direct product of C15 and C4⋊C8

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

Aliases: C15×C4⋊C8, C4⋊C120, C123C40, C6011C8, C205C24, 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

C1C2 — C15×C4⋊C8
C1C2C4C2×C4C2×C20C2×C60C2×C120 — C15×C4⋊C8
C1C2 — C15×C4⋊C8
C1C2×C60 — C15×C4⋊C8

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 [×3], C3, C4 [×2], C4 [×2], C4, C22, C5, C6 [×3], C8 [×2], C2×C4 [×3], C10 [×3], C12 [×2], C12 [×2], C12, C2×C6, C15, C42, C2×C8 [×2], C20 [×2], C20 [×2], C20, C2×C10, C24 [×2], C2×C12 [×3], C30 [×3], C4⋊C8, C40 [×2], C2×C20 [×3], C4×C12, C2×C24 [×2], C60 [×2], C60 [×2], C60, C2×C30, C4×C20, C2×C40 [×2], C3×C4⋊C8, C120 [×2], C2×C60 [×3], C5×C4⋊C8, C4×C60, C2×C120 [×2], C15×C4⋊C8
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C5, C6 [×3], C8 [×2], C2×C4, D4, Q8, C10 [×3], C12 [×2], C2×C6, C15, C4⋊C4, C2×C8, M4(2), C20 [×2], C2×C10, C24 [×2], C2×C12, C3×D4, C3×Q8, C30 [×3], C4⋊C8, C40 [×2], C2×C20, C5×D4, C5×Q8, C3×C4⋊C4, C2×C24, C3×M4(2), C60 [×2], C2×C30, C5×C4⋊C4, C2×C40, C5×M4(2), C3×C4⋊C8, C120 [×2], C2×C60, D4×C15, Q8×C15, C5×C4⋊C8, C15×C4⋊C4, C2×C120, C15×M4(2), C15×C4⋊C8

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

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

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

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

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

dim11111111111111111111222222222222
type++++-
imageC1C2C2C3C4C5C6C6C8C10C10C12C15C20C24C30C30C40C60C120D4Q8M4(2)C3×D4C3×Q8C5×D4C5×Q8C3×M4(2)C5×M4(2)D4×C15Q8×C15C15×M4(2)
kernelC15×C4⋊C8C4×C60C2×C120C5×C4⋊C8C2×C60C3×C4⋊C8C4×C20C2×C40C60C4×C12C2×C24C2×C20C4⋊C8C2×C12C20C42C2×C8C12C2×C4C4C60C60C30C20C20C12C12C10C6C4C4C2
# reps112244248488816168163232641122244488816

Matrix representation of C15×C4⋊C8 in GL4(𝔽241) generated by

225000
09100
0010
0001
,
1000
024000
002402
002401
,
233000
017700
009543
00237146
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

׿
×
𝔽