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## G = C15×C2.D8order 480 = 25·3·5

### Direct product of C15 and C2.D8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C15×C2.D8
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — C2×C60 — C15×C4⋊C4 — C15×C2.D8
 Lower central C1 — C2 — C4 — C15×C2.D8
 Upper central C1 — C2×C30 — C2×C60 — C15×C2.D8

Generators and relations for C15×C2.D8
G = < a,b,c,d | a15=b2=c8=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 104 in 72 conjugacy classes, 56 normal (40 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×2], C22, C5, C6 [×3], C8 [×2], C2×C4, C2×C4 [×2], C10 [×3], C12 [×2], C12 [×2], C2×C6, C15, C4⋊C4 [×2], C2×C8, C20 [×2], C20 [×2], C2×C10, C24 [×2], C2×C12, C2×C12 [×2], C30 [×3], C2.D8, C40 [×2], C2×C20, C2×C20 [×2], C3×C4⋊C4 [×2], C2×C24, C60 [×2], C60 [×2], C2×C30, C5×C4⋊C4 [×2], C2×C40, C3×C2.D8, C120 [×2], C2×C60, C2×C60 [×2], C5×C2.D8, C15×C4⋊C4 [×2], C2×C120, C15×C2.D8
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C5, C6 [×3], C2×C4, D4, Q8, C10 [×3], C12 [×2], C2×C6, C15, C4⋊C4, D8, Q16, C20 [×2], C2×C10, C2×C12, C3×D4, C3×Q8, C30 [×3], C2.D8, C2×C20, C5×D4, C5×Q8, C3×C4⋊C4, C3×D8, C3×Q16, C60 [×2], C2×C30, C5×C4⋊C4, C5×D8, C5×Q16, C3×C2.D8, C2×C60, D4×C15, Q8×C15, C5×C2.D8, C15×C4⋊C4, C15×D8, C15×Q16, C15×C2.D8

Smallest permutation representation of C15×C2.D8
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)
(1 261)(2 262)(3 263)(4 264)(5 265)(6 266)(7 267)(8 268)(9 269)(10 270)(11 256)(12 257)(13 258)(14 259)(15 260)(16 382)(17 383)(18 384)(19 385)(20 386)(21 387)(22 388)(23 389)(24 390)(25 376)(26 377)(27 378)(28 379)(29 380)(30 381)(31 240)(32 226)(33 227)(34 228)(35 229)(36 230)(37 231)(38 232)(39 233)(40 234)(41 235)(42 236)(43 237)(44 238)(45 239)(46 434)(47 435)(48 421)(49 422)(50 423)(51 424)(52 425)(53 426)(54 427)(55 428)(56 429)(57 430)(58 431)(59 432)(60 433)(61 446)(62 447)(63 448)(64 449)(65 450)(66 436)(67 437)(68 438)(69 439)(70 440)(71 441)(72 442)(73 443)(74 444)(75 445)(76 216)(77 217)(78 218)(79 219)(80 220)(81 221)(82 222)(83 223)(84 224)(85 225)(86 211)(87 212)(88 213)(89 214)(90 215)(91 128)(92 129)(93 130)(94 131)(95 132)(96 133)(97 134)(98 135)(99 121)(100 122)(101 123)(102 124)(103 125)(104 126)(105 127)(106 328)(107 329)(108 330)(109 316)(110 317)(111 318)(112 319)(113 320)(114 321)(115 322)(116 323)(117 324)(118 325)(119 326)(120 327)(136 315)(137 301)(138 302)(139 303)(140 304)(141 305)(142 306)(143 307)(144 308)(145 309)(146 310)(147 311)(148 312)(149 313)(150 314)(151 465)(152 451)(153 452)(154 453)(155 454)(156 455)(157 456)(158 457)(159 458)(160 459)(161 460)(162 461)(163 462)(164 463)(165 464)(166 287)(167 288)(168 289)(169 290)(170 291)(171 292)(172 293)(173 294)(174 295)(175 296)(176 297)(177 298)(178 299)(179 300)(180 286)(181 357)(182 358)(183 359)(184 360)(185 346)(186 347)(187 348)(188 349)(189 350)(190 351)(191 352)(192 353)(193 354)(194 355)(195 356)(196 284)(197 285)(198 271)(199 272)(200 273)(201 274)(202 275)(203 276)(204 277)(205 278)(206 279)(207 280)(208 281)(209 282)(210 283)(241 403)(242 404)(243 405)(244 391)(245 392)(246 393)(247 394)(248 395)(249 396)(250 397)(251 398)(252 399)(253 400)(254 401)(255 402)(331 414)(332 415)(333 416)(334 417)(335 418)(336 419)(337 420)(338 406)(339 407)(340 408)(341 409)(342 410)(343 411)(344 412)(345 413)(361 474)(362 475)(363 476)(364 477)(365 478)(366 479)(367 480)(368 466)(369 467)(370 468)(371 469)(372 470)(373 471)(374 472)(375 473)
(1 394 130 136 176 64 158 468)(2 395 131 137 177 65 159 469)(3 396 132 138 178 66 160 470)(4 397 133 139 179 67 161 471)(5 398 134 140 180 68 162 472)(6 399 135 141 166 69 163 473)(7 400 121 142 167 70 164 474)(8 401 122 143 168 71 165 475)(9 402 123 144 169 72 151 476)(10 403 124 145 170 73 152 477)(11 404 125 146 171 74 153 478)(12 405 126 147 172 75 154 479)(13 391 127 148 173 61 155 480)(14 392 128 149 174 62 156 466)(15 393 129 150 175 63 157 467)(16 347 86 113 345 227 271 49)(17 348 87 114 331 228 272 50)(18 349 88 115 332 229 273 51)(19 350 89 116 333 230 274 52)(20 351 90 117 334 231 275 53)(21 352 76 118 335 232 276 54)(22 353 77 119 336 233 277 55)(23 354 78 120 337 234 278 56)(24 355 79 106 338 235 279 57)(25 356 80 107 339 236 280 58)(26 357 81 108 340 237 281 59)(27 358 82 109 341 238 282 60)(28 359 83 110 342 239 283 46)(29 360 84 111 343 240 284 47)(30 346 85 112 344 226 285 48)(31 196 435 380 184 224 318 411)(32 197 421 381 185 225 319 412)(33 198 422 382 186 211 320 413)(34 199 423 383 187 212 321 414)(35 200 424 384 188 213 322 415)(36 201 425 385 189 214 323 416)(37 202 426 386 190 215 324 417)(38 203 427 387 191 216 325 418)(39 204 428 388 192 217 326 419)(40 205 429 389 193 218 327 420)(41 206 430 390 194 219 328 406)(42 207 431 376 195 220 329 407)(43 208 432 377 181 221 330 408)(44 209 433 378 182 222 316 409)(45 210 434 379 183 223 317 410)(91 313 295 447 455 368 259 245)(92 314 296 448 456 369 260 246)(93 315 297 449 457 370 261 247)(94 301 298 450 458 371 262 248)(95 302 299 436 459 372 263 249)(96 303 300 437 460 373 264 250)(97 304 286 438 461 374 265 251)(98 305 287 439 462 375 266 252)(99 306 288 440 463 361 267 253)(100 307 289 441 464 362 268 254)(101 308 290 442 465 363 269 255)(102 309 291 443 451 364 270 241)(103 310 292 444 452 365 256 242)(104 311 293 445 453 366 257 243)(105 312 294 446 454 367 258 244)
(1 197 261 285)(2 198 262 271)(3 199 263 272)(4 200 264 273)(5 201 265 274)(6 202 266 275)(7 203 267 276)(8 204 268 277)(9 205 269 278)(10 206 270 279)(11 207 256 280)(12 208 257 281)(13 209 258 282)(14 210 259 283)(15 196 260 284)(16 159 382 458)(17 160 383 459)(18 161 384 460)(19 162 385 461)(20 163 386 462)(21 164 387 463)(22 165 388 464)(23 151 389 465)(24 152 390 451)(25 153 376 452)(26 154 377 453)(27 155 378 454)(28 156 379 455)(29 157 380 456)(30 158 381 457)(31 246 240 393)(32 247 226 394)(33 248 227 395)(34 249 228 396)(35 250 229 397)(36 251 230 398)(37 252 231 399)(38 253 232 400)(39 254 233 401)(40 255 234 402)(41 241 235 403)(42 242 236 404)(43 243 237 405)(44 244 238 391)(45 245 239 392)(46 466 434 368)(47 467 435 369)(48 468 421 370)(49 469 422 371)(50 470 423 372)(51 471 424 373)(52 472 425 374)(53 473 426 375)(54 474 427 361)(55 475 428 362)(56 476 429 363)(57 477 430 364)(58 478 431 365)(59 479 432 366)(60 480 433 367)(61 182 446 358)(62 183 447 359)(63 184 448 360)(64 185 449 346)(65 186 450 347)(66 187 436 348)(67 188 437 349)(68 189 438 350)(69 190 439 351)(70 191 440 352)(71 192 441 353)(72 193 442 354)(73 194 443 355)(74 195 444 356)(75 181 445 357)(76 167 216 288)(77 168 217 289)(78 169 218 290)(79 170 219 291)(80 171 220 292)(81 172 221 293)(82 173 222 294)(83 174 223 295)(84 175 224 296)(85 176 225 297)(86 177 211 298)(87 178 212 299)(88 179 213 300)(89 180 214 286)(90 166 215 287)(91 342 128 410)(92 343 129 411)(93 344 130 412)(94 345 131 413)(95 331 132 414)(96 332 133 415)(97 333 134 416)(98 334 135 417)(99 335 121 418)(100 336 122 419)(101 337 123 420)(102 338 124 406)(103 339 125 407)(104 340 126 408)(105 341 127 409)(106 145 328 309)(107 146 329 310)(108 147 330 311)(109 148 316 312)(110 149 317 313)(111 150 318 314)(112 136 319 315)(113 137 320 301)(114 138 321 302)(115 139 322 303)(116 140 323 304)(117 141 324 305)(118 142 325 306)(119 143 326 307)(120 144 327 308)

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

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

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

210 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 6A ··· 6F 8A 8B 8C 8D 10A ··· 10L 12A 12B 12C 12D 12E ··· 12L 15A ··· 15H 20A ··· 20H 20I ··· 20X 24A ··· 24H 30A ··· 30X 40A ··· 40P 60A ··· 60P 60Q ··· 60AV 120A ··· 120AF order 1 2 2 2 3 3 4 4 4 4 4 4 5 5 5 5 6 ··· 6 8 8 8 8 10 ··· 10 12 12 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 2 2 4 4 4 4 1 1 1 1 1 ··· 1 2 2 2 2 1 ··· 1 2 2 2 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2

210 irreducible representations

 dim 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 2 2 2 2 type + + + - + + - image C1 C2 C2 C3 C4 C5 C6 C6 C10 C10 C12 C15 C20 C30 C30 C60 Q8 D4 D8 Q16 C3×Q8 C3×D4 C5×Q8 C5×D4 C3×D8 C3×Q16 C5×D8 C5×Q16 Q8×C15 D4×C15 C15×D8 C15×Q16 kernel C15×C2.D8 C15×C4⋊C4 C2×C120 C5×C2.D8 C120 C3×C2.D8 C5×C4⋊C4 C2×C40 C3×C4⋊C4 C2×C24 C40 C2.D8 C24 C4⋊C4 C2×C8 C8 C60 C2×C30 C30 C30 C20 C2×C10 C12 C2×C6 C10 C10 C6 C6 C4 C22 C2 C2 # reps 1 2 1 2 4 4 4 2 8 4 8 8 16 16 8 32 1 1 2 2 2 2 4 4 4 4 8 8 8 8 16 16

Matrix representation of C15×C2.D8 in GL3(𝔽241) generated by

 1 0 0 0 100 0 0 0 100
,
 240 0 0 0 240 0 0 0 240
,
 1 0 0 0 0 219 0 11 219
,
 64 0 0 0 104 64 0 136 137
G:=sub<GL(3,GF(241))| [1,0,0,0,100,0,0,0,100],[240,0,0,0,240,0,0,0,240],[1,0,0,0,0,11,0,219,219],[64,0,0,0,104,136,0,64,137] >;

C15×C2.D8 in GAP, Magma, Sage, TeX

C_{15}\times C_2.D_8
% in TeX

G:=Group("C15xC2.D8");
// GroupNames label

G:=SmallGroup(480,210);
// by ID

G=gap.SmallGroup(480,210);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,840,869,2108,10504,172]);
// Polycyclic

G:=Group<a,b,c,d|a^15=b^2=c^8=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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