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G = C15×Q8⋊C4order 480 = 25·3·5

Direct product of C15 and Q8⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C15×Q8⋊C4, Q82C60, C60.246D4, C30.25Q16, C30.41SD16, C4⋊C4.1C30, C4.2(C2×C60), (C2×C40).3C6, (C2×C8).1C30, (C3×Q8)⋊4C20, C6.6(C5×Q16), (C2×C24).3C10, (C2×C120).7C2, (C5×Q8)⋊10C12, (Q8×C15)⋊16C4, C12.61(C5×D4), C20.61(C3×D4), C4.12(D4×C15), (C2×Q8).4C30, (C6×Q8).7C10, C2.1(C15×Q16), C10.6(C3×Q16), C12.30(C2×C20), C20.51(C2×C12), C60.226(C2×C4), (C2×C30).190D4, (Q8×C10).11C6, (Q8×C30).17C2, C2.2(C15×SD16), C6.10(C5×SD16), C22.9(D4×C15), C10.10(C3×SD16), (C2×C60).570C22, C30.129(C22⋊C4), (C5×C4⋊C4).8C6, (C3×C4⋊C4).8C10, (C2×C6).47(C5×D4), (C15×C4⋊C4).22C2, (C2×C4).17(C2×C30), (C2×C10).47(C3×D4), C6.25(C5×C22⋊C4), C2.7(C15×C22⋊C4), (C2×C20).118(C2×C6), C10.36(C3×C22⋊C4), (C2×C12).121(C2×C10), SmallGroup(480,206)

Series: Derived Chief Lower central Upper central

C1C4 — C15×Q8⋊C4
C1C2C22C2×C4C2×C20C2×C60C15×C4⋊C4 — C15×Q8⋊C4
C1C2C4 — C15×Q8⋊C4
C1C2×C30C2×C60 — C15×Q8⋊C4

Generators and relations for C15×Q8⋊C4
 G = < a,b,c,d | a15=b4=d4=1, c2=b2, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b-1c >

Subgroups: 120 in 84 conjugacy classes, 56 normal (48 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×3], C22, C5, C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×2], Q8, C10 [×3], C12 [×2], C12 [×3], C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, C20 [×2], C20 [×3], C2×C10, C24, C2×C12, C2×C12 [×2], C3×Q8 [×2], C3×Q8, C30 [×3], Q8⋊C4, C40, C2×C20, C2×C20 [×2], C5×Q8 [×2], C5×Q8, C3×C4⋊C4, C2×C24, C6×Q8, C60 [×2], C60 [×3], C2×C30, C5×C4⋊C4, C2×C40, Q8×C10, C3×Q8⋊C4, C120, C2×C60, C2×C60 [×2], Q8×C15 [×2], Q8×C15, C5×Q8⋊C4, C15×C4⋊C4, C2×C120, Q8×C30, C15×Q8⋊C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C5, C6 [×3], C2×C4, D4 [×2], C10 [×3], C12 [×2], C2×C6, C15, C22⋊C4, SD16, Q16, C20 [×2], C2×C10, C2×C12, C3×D4 [×2], C30 [×3], Q8⋊C4, C2×C20, C5×D4 [×2], C3×C22⋊C4, C3×SD16, C3×Q16, C60 [×2], C2×C30, C5×C22⋊C4, C5×SD16, C5×Q16, C3×Q8⋊C4, C2×C60, D4×C15 [×2], C5×Q8⋊C4, C15×C22⋊C4, C15×SD16, C15×Q16, C15×Q8⋊C4

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

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

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

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

210 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F5A5B5C5D6A···6F8A8B8C8D10A···10L12A12B12C12D12E···12L15A···15H20A···20H20I···20X24A···24H30A···30X40A···40P60A···60P60Q···60AV120A···120AF
order12223344444455556···6888810···101212121212···1215···1520···2020···2024···2430···3040···4060···6060···60120···120
size11111122444411111···122221···122224···41···12···24···42···21···12···22···24···42···2

210 irreducible representations

dim111111111111111111112222222222222222
type++++++-
imageC1C2C2C2C3C4C5C6C6C6C10C10C10C12C15C20C30C30C30C60D4D4SD16Q16C3×D4C3×D4C5×D4C5×D4C3×SD16C3×Q16C5×SD16C5×Q16D4×C15D4×C15C15×SD16C15×Q16
kernelC15×Q8⋊C4C15×C4⋊C4C2×C120Q8×C30C5×Q8⋊C4Q8×C15C3×Q8⋊C4C5×C4⋊C4C2×C40Q8×C10C3×C4⋊C4C2×C24C6×Q8C5×Q8Q8⋊C4C3×Q8C4⋊C4C2×C8C2×Q8Q8C60C2×C30C30C30C20C2×C10C12C2×C6C10C10C6C6C4C22C2C2
# reps1111244222444881688832112222444488881616

Matrix representation of C15×Q8⋊C4 in GL4(𝔽241) generated by

1000
022500
00870
00087
,
1000
0100
0001
002400
,
240000
024000
00358
0058238
,
64000
0100
0052118
00118189
G:=sub<GL(4,GF(241))| [1,0,0,0,0,225,0,0,0,0,87,0,0,0,0,87],[1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,0],[240,0,0,0,0,240,0,0,0,0,3,58,0,0,58,238],[64,0,0,0,0,1,0,0,0,0,52,118,0,0,118,189] >;

C15×Q8⋊C4 in GAP, Magma, Sage, TeX

C_{15}\times Q_8\rtimes C_4
% in TeX

G:=Group("C15xQ8:C4");
// GroupNames label

G:=SmallGroup(480,206);
// by ID

G=gap.SmallGroup(480,206);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,840,869,1688,10504,5261,172]);
// Polycyclic

G:=Group<a,b,c,d|a^15=b^4=d^4=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
// generators/relations

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