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G = C22×C15⋊Q8order 480 = 25·3·5

Direct product of C22 and C15⋊Q8

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22×C15⋊Q8, C30.49C24, Dic15.44C23, C304(C2×Q8), (C2×C30)⋊5Q8, C155(C22×Q8), C102(C2×Dic6), (C2×C10)⋊9Dic6, (C2×C6)⋊6Dic10, C62(C2×Dic10), C52(C22×Dic6), C23.73(S3×D5), C6.49(C23×D5), C32(C22×Dic10), C10.49(S3×C23), (C2×C30).252C23, (C2×Dic5).199D6, (C22×C10).121D6, (C22×C6).104D10, (C22×Dic5).9S3, (C22×Dic3).7D5, (C2×Dic3).170D10, (C22×C30).90C22, Dic3.31(C22×D5), (C5×Dic3).36C23, Dic5.47(C22×S3), (C3×Dic5).52C23, (C22×Dic15).14C2, (C6×Dic5).230C22, (C2×Dic15).235C22, (C10×Dic3).210C22, (C2×C6×Dic5).8C2, C2.49(C22×S3×D5), (Dic3×C2×C10).8C2, C22.112(C2×S3×D5), (C2×C6).258(C22×D5), (C2×C10).256(C22×S3), SmallGroup(480,1121)

Series: Derived Chief Lower central Upper central

C1C30 — C22×C15⋊Q8
C1C5C15C30C3×Dic5C15⋊Q8C2×C15⋊Q8 — C22×C15⋊Q8
C15C30 — C22×C15⋊Q8
C1C23

Generators and relations for C22×C15⋊Q8
 G = < a,b,c,d,e | a2=b2=c15=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c11, ece-1=c4, ede-1=d-1 >

Subgroups: 1244 in 312 conjugacy classes, 148 normal (18 characteristic)
C1, C2, C2 [×6], C3, C4 [×12], C22 [×7], C5, C6, C6 [×6], C2×C4 [×18], Q8 [×16], C23, C10, C10 [×6], Dic3 [×4], Dic3 [×4], C12 [×4], C2×C6 [×7], C15, C22×C4 [×3], C2×Q8 [×12], Dic5 [×4], Dic5 [×4], C20 [×4], C2×C10 [×7], Dic6 [×16], C2×Dic3 [×6], C2×Dic3 [×6], C2×C12 [×6], C22×C6, C30, C30 [×6], C22×Q8, Dic10 [×16], C2×Dic5 [×6], C2×Dic5 [×6], C2×C20 [×6], C22×C10, C2×Dic6 [×12], C22×Dic3, C22×Dic3, C22×C12, C5×Dic3 [×4], C3×Dic5 [×4], Dic15 [×4], C2×C30 [×7], C2×Dic10 [×12], C22×Dic5, C22×Dic5, C22×C20, C22×Dic6, C15⋊Q8 [×16], C6×Dic5 [×6], C10×Dic3 [×6], C2×Dic15 [×6], C22×C30, C22×Dic10, C2×C15⋊Q8 [×12], C2×C6×Dic5, Dic3×C2×C10, C22×Dic15, C22×C15⋊Q8
Quotients: C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D5, D6 [×7], C2×Q8 [×6], C24, D10 [×7], Dic6 [×4], C22×S3 [×7], C22×Q8, Dic10 [×4], C22×D5 [×7], C2×Dic6 [×6], S3×C23, S3×D5, C2×Dic10 [×6], C23×D5, C22×Dic6, C15⋊Q8 [×4], C2×S3×D5 [×3], C22×Dic10, C2×C15⋊Q8 [×6], C22×S3×D5, C22×C15⋊Q8

Smallest permutation representation of C22×C15⋊Q8
Regular action on 480 points
Generators in S480
(1 113)(2 114)(3 115)(4 116)(5 117)(6 118)(7 119)(8 120)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 112)(16 101)(17 102)(18 103)(19 104)(20 105)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 97)(28 98)(29 99)(30 100)(31 81)(32 82)(33 83)(34 84)(35 85)(36 86)(37 87)(38 88)(39 89)(40 90)(41 76)(42 77)(43 78)(44 79)(45 80)(46 63)(47 64)(48 65)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)(55 72)(56 73)(57 74)(58 75)(59 61)(60 62)(121 230)(122 231)(123 232)(124 233)(125 234)(126 235)(127 236)(128 237)(129 238)(130 239)(131 240)(132 226)(133 227)(134 228)(135 229)(136 213)(137 214)(138 215)(139 216)(140 217)(141 218)(142 219)(143 220)(144 221)(145 222)(146 223)(147 224)(148 225)(149 211)(150 212)(151 198)(152 199)(153 200)(154 201)(155 202)(156 203)(157 204)(158 205)(159 206)(160 207)(161 208)(162 209)(163 210)(164 196)(165 197)(166 181)(167 182)(168 183)(169 184)(170 185)(171 186)(172 187)(173 188)(174 189)(175 190)(176 191)(177 192)(178 193)(179 194)(180 195)(241 359)(242 360)(243 346)(244 347)(245 348)(246 349)(247 350)(248 351)(249 352)(250 353)(251 354)(252 355)(253 356)(254 357)(255 358)(256 331)(257 332)(258 333)(259 334)(260 335)(261 336)(262 337)(263 338)(264 339)(265 340)(266 341)(267 342)(268 343)(269 344)(270 345)(271 328)(272 329)(273 330)(274 316)(275 317)(276 318)(277 319)(278 320)(279 321)(280 322)(281 323)(282 324)(283 325)(284 326)(285 327)(286 308)(287 309)(288 310)(289 311)(290 312)(291 313)(292 314)(293 315)(294 301)(295 302)(296 303)(297 304)(298 305)(299 306)(300 307)(361 470)(362 471)(363 472)(364 473)(365 474)(366 475)(367 476)(368 477)(369 478)(370 479)(371 480)(372 466)(373 467)(374 468)(375 469)(376 455)(377 456)(378 457)(379 458)(380 459)(381 460)(382 461)(383 462)(384 463)(385 464)(386 465)(387 451)(388 452)(389 453)(390 454)(391 440)(392 441)(393 442)(394 443)(395 444)(396 445)(397 446)(398 447)(399 448)(400 449)(401 450)(402 436)(403 437)(404 438)(405 439)(406 424)(407 425)(408 426)(409 427)(410 428)(411 429)(412 430)(413 431)(414 432)(415 433)(416 434)(417 435)(418 421)(419 422)(420 423)
(1 59)(2 60)(3 46)(4 47)(5 48)(6 49)(7 50)(8 51)(9 52)(10 53)(11 54)(12 55)(13 56)(14 57)(15 58)(16 32)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)(29 45)(30 31)(61 113)(62 114)(63 115)(64 116)(65 117)(66 118)(67 119)(68 120)(69 106)(70 107)(71 108)(72 109)(73 110)(74 111)(75 112)(76 95)(77 96)(78 97)(79 98)(80 99)(81 100)(82 101)(83 102)(84 103)(85 104)(86 105)(87 91)(88 92)(89 93)(90 94)(121 170)(122 171)(123 172)(124 173)(125 174)(126 175)(127 176)(128 177)(129 178)(130 179)(131 180)(132 166)(133 167)(134 168)(135 169)(136 164)(137 165)(138 151)(139 152)(140 153)(141 154)(142 155)(143 156)(144 157)(145 158)(146 159)(147 160)(148 161)(149 162)(150 163)(181 226)(182 227)(183 228)(184 229)(185 230)(186 231)(187 232)(188 233)(189 234)(190 235)(191 236)(192 237)(193 238)(194 239)(195 240)(196 213)(197 214)(198 215)(199 216)(200 217)(201 218)(202 219)(203 220)(204 221)(205 222)(206 223)(207 224)(208 225)(209 211)(210 212)(241 286)(242 287)(243 288)(244 289)(245 290)(246 291)(247 292)(248 293)(249 294)(250 295)(251 296)(252 297)(253 298)(254 299)(255 300)(256 277)(257 278)(258 279)(259 280)(260 281)(261 282)(262 283)(263 284)(264 285)(265 271)(266 272)(267 273)(268 274)(269 275)(270 276)(301 352)(302 353)(303 354)(304 355)(305 356)(306 357)(307 358)(308 359)(309 360)(310 346)(311 347)(312 348)(313 349)(314 350)(315 351)(316 343)(317 344)(318 345)(319 331)(320 332)(321 333)(322 334)(323 335)(324 336)(325 337)(326 338)(327 339)(328 340)(329 341)(330 342)(361 407)(362 408)(363 409)(364 410)(365 411)(366 412)(367 413)(368 414)(369 415)(370 416)(371 417)(372 418)(373 419)(374 420)(375 406)(376 399)(377 400)(378 401)(379 402)(380 403)(381 404)(382 405)(383 391)(384 392)(385 393)(386 394)(387 395)(388 396)(389 397)(390 398)(421 466)(422 467)(423 468)(424 469)(425 470)(426 471)(427 472)(428 473)(429 474)(430 475)(431 476)(432 477)(433 478)(434 479)(435 480)(436 458)(437 459)(438 460)(439 461)(440 462)(441 463)(442 464)(443 465)(444 451)(445 452)(446 453)(447 454)(448 455)(449 456)(450 457)
(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)
(1 226 21 222)(2 237 22 218)(3 233 23 214)(4 229 24 225)(5 240 25 221)(6 236 26 217)(7 232 27 213)(8 228 28 224)(9 239 29 220)(10 235 30 216)(11 231 16 212)(12 227 17 223)(13 238 18 219)(14 234 19 215)(15 230 20 211)(31 199 53 190)(32 210 54 186)(33 206 55 182)(34 202 56 193)(35 198 57 189)(36 209 58 185)(37 205 59 181)(38 201 60 192)(39 197 46 188)(40 208 47 184)(41 204 48 195)(42 200 49 191)(43 196 50 187)(44 207 51 183)(45 203 52 194)(61 166 87 158)(62 177 88 154)(63 173 89 165)(64 169 90 161)(65 180 76 157)(66 176 77 153)(67 172 78 164)(68 168 79 160)(69 179 80 156)(70 175 81 152)(71 171 82 163)(72 167 83 159)(73 178 84 155)(74 174 85 151)(75 170 86 162)(91 145 113 132)(92 141 114 128)(93 137 115 124)(94 148 116 135)(95 144 117 131)(96 140 118 127)(97 136 119 123)(98 147 120 134)(99 143 106 130)(100 139 107 126)(101 150 108 122)(102 146 109 133)(103 142 110 129)(104 138 111 125)(105 149 112 121)(241 456 257 467)(242 452 258 478)(243 463 259 474)(244 459 260 470)(245 455 261 466)(246 451 262 477)(247 462 263 473)(248 458 264 469)(249 454 265 480)(250 465 266 476)(251 461 267 472)(252 457 268 468)(253 453 269 479)(254 464 270 475)(255 460 256 471)(271 435 294 447)(272 431 295 443)(273 427 296 439)(274 423 297 450)(275 434 298 446)(276 430 299 442)(277 426 300 438)(278 422 286 449)(279 433 287 445)(280 429 288 441)(281 425 289 437)(282 421 290 448)(283 432 291 444)(284 428 292 440)(285 424 293 436)(301 398 328 417)(302 394 329 413)(303 405 330 409)(304 401 316 420)(305 397 317 416)(306 393 318 412)(307 404 319 408)(308 400 320 419)(309 396 321 415)(310 392 322 411)(311 403 323 407)(312 399 324 418)(313 395 325 414)(314 391 326 410)(315 402 327 406)(331 362 358 381)(332 373 359 377)(333 369 360 388)(334 365 346 384)(335 361 347 380)(336 372 348 376)(337 368 349 387)(338 364 350 383)(339 375 351 379)(340 371 352 390)(341 367 353 386)(342 363 354 382)(343 374 355 378)(344 370 356 389)(345 366 357 385)
(1 347 21 335)(2 351 22 339)(3 355 23 343)(4 359 24 332)(5 348 25 336)(6 352 26 340)(7 356 27 344)(8 360 28 333)(9 349 29 337)(10 353 30 341)(11 357 16 345)(12 346 17 334)(13 350 18 338)(14 354 19 342)(15 358 20 331)(31 329 53 302)(32 318 54 306)(33 322 55 310)(34 326 56 314)(35 330 57 303)(36 319 58 307)(37 323 59 311)(38 327 60 315)(39 316 46 304)(40 320 47 308)(41 324 48 312)(42 328 49 301)(43 317 50 305)(44 321 51 309)(45 325 52 313)(61 289 87 281)(62 293 88 285)(63 297 89 274)(64 286 90 278)(65 290 76 282)(66 294 77 271)(67 298 78 275)(68 287 79 279)(69 291 80 283)(70 295 81 272)(71 299 82 276)(72 288 83 280)(73 292 84 284)(74 296 85 273)(75 300 86 277)(91 260 113 244)(92 264 114 248)(93 268 115 252)(94 257 116 241)(95 261 117 245)(96 265 118 249)(97 269 119 253)(98 258 120 242)(99 262 106 246)(100 266 107 250)(101 270 108 254)(102 259 109 243)(103 263 110 247)(104 267 111 251)(105 256 112 255)(121 471 149 460)(122 475 150 464)(123 479 136 453)(124 468 137 457)(125 472 138 461)(126 476 139 465)(127 480 140 454)(128 469 141 458)(129 473 142 462)(130 477 143 451)(131 466 144 455)(132 470 145 459)(133 474 146 463)(134 478 147 452)(135 467 148 456)(151 439 174 427)(152 443 175 431)(153 447 176 435)(154 436 177 424)(155 440 178 428)(156 444 179 432)(157 448 180 421)(158 437 166 425)(159 441 167 429)(160 445 168 433)(161 449 169 422)(162 438 170 426)(163 442 171 430)(164 446 172 434)(165 450 173 423)(181 407 205 403)(182 411 206 392)(183 415 207 396)(184 419 208 400)(185 408 209 404)(186 412 210 393)(187 416 196 397)(188 420 197 401)(189 409 198 405)(190 413 199 394)(191 417 200 398)(192 406 201 402)(193 410 202 391)(194 414 203 395)(195 418 204 399)(211 381 230 362)(212 385 231 366)(213 389 232 370)(214 378 233 374)(215 382 234 363)(216 386 235 367)(217 390 236 371)(218 379 237 375)(219 383 238 364)(220 387 239 368)(221 376 240 372)(222 380 226 361)(223 384 227 365)(224 388 228 369)(225 377 229 373)

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

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

84 conjugacy classes

class 1 2A···2G 3 4A4B4C4D4E4F4G4H4I4J4K4L5A5B6A···6G10A···10N12A···12H15A15B20A···20P30A···30N
order12···23444444444444556···610···1012···12151520···2030···30
size11···1266661010101030303030222···22···210···10446···64···4

84 irreducible representations

dim11111222222222444
type++++++-+++++--+-+
imageC1C2C2C2C2S3Q8D5D6D6D10D10Dic6Dic10S3×D5C15⋊Q8C2×S3×D5
kernelC22×C15⋊Q8C2×C15⋊Q8C2×C6×Dic5Dic3×C2×C10C22×Dic15C22×Dic5C2×C30C22×Dic3C2×Dic5C22×C10C2×Dic3C22×C6C2×C10C2×C6C23C22C22
# reps11211114261122816286

Matrix representation of C22×C15⋊Q8 in GL5(𝔽61)

10000
060000
006000
000600
000060
,
600000
01000
00100
00010
00001
,
10000
0606000
01000
000044
0001843
,
10000
0221300
0523900
000257
0005036
,
600000
060000
006000
0003114
000130

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,60,1,0,0,0,60,0,0,0,0,0,0,0,18,0,0,0,44,43],[1,0,0,0,0,0,22,52,0,0,0,13,39,0,0,0,0,0,25,50,0,0,0,7,36],[60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,31,1,0,0,0,14,30] >;

C22×C15⋊Q8 in GAP, Magma, Sage, TeX

C_2^2\times C_{15}\rtimes Q_8
% in TeX

G:=Group("C2^2xC15:Q8");
// GroupNames label

G:=SmallGroup(480,1121);
// by ID

G=gap.SmallGroup(480,1121);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,253,120,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^15=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^11,e*c*e^-1=c^4,e*d*e^-1=d^-1>;
// generators/relations

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