direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C23×Dic15, C24.3D15, C30.67C24, C23.41D30, C15⋊11(C23×C4), (C23×C6).3D5, C3⋊2(C23×Dic5), C5⋊3(C23×Dic3), (C22×C30)⋊13C4, C30⋊11(C22×C4), (C23×C10).5S3, (C23×C30).3C2, (C22×C6)⋊5Dic5, C6⋊2(C22×Dic5), C6.67(C23×D5), C2.2(C23×D15), C10.67(S3×C23), C10⋊3(C22×Dic3), (C2×C30).325C23, (C22×C10)⋊11Dic3, (C22×C6).128D10, (C22×C10).146D6, C22.33(C22×D15), (C22×C30).151C22, (C2×C30)⋊41(C2×C4), (C2×C6)⋊9(C2×Dic5), (C2×C10)⋊16(C2×Dic3), (C2×C6).321(C22×D5), (C2×C10).321(C22×S3), SmallGroup(480,1178)
Series: Derived ►Chief ►Lower central ►Upper central
| C15 — C23×Dic15 |
Generators and relations for C23×Dic15
G = < a,b,c,d,e | a2=b2=c2=d30=1, e2=d15, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 1492 in 472 conjugacy classes, 319 normal (13 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C2×C4, C23, C10, C10, Dic3, C2×C6, C15, C22×C4, C24, Dic5, C2×C10, C2×Dic3, C22×C6, C30, C30, C23×C4, C2×Dic5, C22×C10, C22×Dic3, C23×C6, Dic15, C2×C30, C22×Dic5, C23×C10, C23×Dic3, C2×Dic15, C22×C30, C23×Dic5, C22×Dic15, C23×C30, C23×Dic15
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, Dic3, D6, C22×C4, C24, Dic5, D10, C2×Dic3, C22×S3, D15, C23×C4, C2×Dic5, C22×D5, C22×Dic3, S3×C23, Dic15, D30, C22×Dic5, C23×D5, C23×Dic3, C2×Dic15, C22×D15, C23×Dic5, C22×Dic15, C23×D15, C23×Dic15
►Regular action on 480 points
Generators in S480
(1 121)(2 122)(3 123)(4 124)(5 125)(6 126)(7 127)(8 128)(9 129)(10 130)(11 131)(12 132)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(21 141)(22 142)(23 143)(24 144)(25 145)(26 146)(27 147)(28 148)(29 149)(30 150)(31 166)(32 167)(33 168)(34 169)(35 170)(36 171)(37 172)(38 173)(39 174)(40 175)(41 176)(42 177)(43 178)(44 179)(45 180)(46 151)(47 152)(48 153)(49 154)(50 155)(51 156)(52 157)(53 158)(54 159)(55 160)(56 161)(57 162)(58 163)(59 164)(60 165)(61 196)(62 197)(63 198)(64 199)(65 200)(66 201)(67 202)(68 203)(69 204)(70 205)(71 206)(72 207)(73 208)(74 209)(75 210)(76 181)(77 182)(78 183)(79 184)(80 185)(81 186)(82 187)(83 188)(84 189)(85 190)(86 191)(87 192)(88 193)(89 194)(90 195)(91 226)(92 227)(93 228)(94 229)(95 230)(96 231)(97 232)(98 233)(99 234)(100 235)(101 236)(102 237)(103 238)(104 239)(105 240)(106 211)(107 212)(108 213)(109 214)(110 215)(111 216)(112 217)(113 218)(114 219)(115 220)(116 221)(117 222)(118 223)(119 224)(120 225)(241 371)(242 372)(243 373)(244 374)(245 375)(246 376)(247 377)(248 378)(249 379)(250 380)(251 381)(252 382)(253 383)(254 384)(255 385)(256 386)(257 387)(258 388)(259 389)(260 390)(261 361)(262 362)(263 363)(264 364)(265 365)(266 366)(267 367)(268 368)(269 369)(270 370)(271 411)(272 412)(273 413)(274 414)(275 415)(276 416)(277 417)(278 418)(279 419)(280 420)(281 391)(282 392)(283 393)(284 394)(285 395)(286 396)(287 397)(288 398)(289 399)(290 400)(291 401)(292 402)(293 403)(294 404)(295 405)(296 406)(297 407)(298 408)(299 409)(300 410)(301 446)(302 447)(303 448)(304 449)(305 450)(306 421)(307 422)(308 423)(309 424)(310 425)(311 426)(312 427)(313 428)(314 429)(315 430)(316 431)(317 432)(318 433)(319 434)(320 435)(321 436)(322 437)(323 438)(324 439)(325 440)(326 441)(327 442)(328 443)(329 444)(330 445)(331 476)(332 477)(333 478)(334 479)(335 480)(336 451)(337 452)(338 453)(339 454)(340 455)(341 456)(342 457)(343 458)(344 459)(345 460)(346 461)(347 462)(348 463)(349 464)(350 465)(351 466)(352 467)(353 468)(354 469)(355 470)(356 471)(357 472)(358 473)(359 474)(360 475)
(1 106)(2 107)(3 108)(4 109)(5 110)(6 111)(7 112)(8 113)(9 114)(10 115)(11 116)(12 117)(13 118)(14 119)(15 120)(16 91)(17 92)(18 93)(19 94)(20 95)(21 96)(22 97)(23 98)(24 99)(25 100)(26 101)(27 102)(28 103)(29 104)(30 105)(31 69)(32 70)(33 71)(34 72)(35 73)(36 74)(37 75)(38 76)(39 77)(40 78)(41 79)(42 80)(43 81)(44 82)(45 83)(46 84)(47 85)(48 86)(49 87)(50 88)(51 89)(52 90)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)(121 211)(122 212)(123 213)(124 214)(125 215)(126 216)(127 217)(128 218)(129 219)(130 220)(131 221)(132 222)(133 223)(134 224)(135 225)(136 226)(137 227)(138 228)(139 229)(140 230)(141 231)(142 232)(143 233)(144 234)(145 235)(146 236)(147 237)(148 238)(149 239)(150 240)(151 189)(152 190)(153 191)(154 192)(155 193)(156 194)(157 195)(158 196)(159 197)(160 198)(161 199)(162 200)(163 201)(164 202)(165 203)(166 204)(167 205)(168 206)(169 207)(170 208)(171 209)(172 210)(173 181)(174 182)(175 183)(176 184)(177 185)(178 186)(179 187)(180 188)(241 346)(242 347)(243 348)(244 349)(245 350)(246 351)(247 352)(248 353)(249 354)(250 355)(251 356)(252 357)(253 358)(254 359)(255 360)(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)(271 321)(272 322)(273 323)(274 324)(275 325)(276 326)(277 327)(278 328)(279 329)(280 330)(281 301)(282 302)(283 303)(284 304)(285 305)(286 306)(287 307)(288 308)(289 309)(290 310)(291 311)(292 312)(293 313)(294 314)(295 315)(296 316)(297 317)(298 318)(299 319)(300 320)(361 451)(362 452)(363 453)(364 454)(365 455)(366 456)(367 457)(368 458)(369 459)(370 460)(371 461)(372 462)(373 463)(374 464)(375 465)(376 466)(377 467)(378 468)(379 469)(380 470)(381 471)(382 472)(383 473)(384 474)(385 475)(386 476)(387 477)(388 478)(389 479)(390 480)(391 446)(392 447)(393 448)(394 449)(395 450)(396 421)(397 422)(398 423)(399 424)(400 425)(401 426)(402 427)(403 428)(404 429)(405 430)(406 431)(407 432)(408 433)(409 434)(410 435)(411 436)(412 437)(413 438)(414 439)(415 440)(416 441)(417 442)(418 443)(419 444)(420 445)
(1 59)(2 60)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(61 100)(62 101)(63 102)(64 103)(65 104)(66 105)(67 106)(68 107)(69 108)(70 109)(71 110)(72 111)(73 112)(74 113)(75 114)(76 115)(77 116)(78 117)(79 118)(80 119)(81 120)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 97)(89 98)(90 99)(121 164)(122 165)(123 166)(124 167)(125 168)(126 169)(127 170)(128 171)(129 172)(130 173)(131 174)(132 175)(133 176)(134 177)(135 178)(136 179)(137 180)(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)(181 220)(182 221)(183 222)(184 223)(185 224)(186 225)(187 226)(188 227)(189 228)(190 229)(191 230)(192 231)(193 232)(194 233)(195 234)(196 235)(197 236)(198 237)(199 238)(200 239)(201 240)(202 211)(203 212)(204 213)(205 214)(206 215)(207 216)(208 217)(209 218)(210 219)(241 281)(242 282)(243 283)(244 284)(245 285)(246 286)(247 287)(248 288)(249 289)(250 290)(251 291)(252 292)(253 293)(254 294)(255 295)(256 296)(257 297)(258 298)(259 299)(260 300)(261 271)(262 272)(263 273)(264 274)(265 275)(266 276)(267 277)(268 278)(269 279)(270 280)(301 346)(302 347)(303 348)(304 349)(305 350)(306 351)(307 352)(308 353)(309 354)(310 355)(311 356)(312 357)(313 358)(314 359)(315 360)(316 331)(317 332)(318 333)(319 334)(320 335)(321 336)(322 337)(323 338)(324 339)(325 340)(326 341)(327 342)(328 343)(329 344)(330 345)(361 411)(362 412)(363 413)(364 414)(365 415)(366 416)(367 417)(368 418)(369 419)(370 420)(371 391)(372 392)(373 393)(374 394)(375 395)(376 396)(377 397)(378 398)(379 399)(380 400)(381 401)(382 402)(383 403)(384 404)(385 405)(386 406)(387 407)(388 408)(389 409)(390 410)(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)(436 451)(437 452)(438 453)(439 454)(440 455)(441 456)(442 457)(443 458)(444 459)(445 460)(446 461)(447 462)(448 463)(449 464)(450 465)
(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)
(1 426 16 441)(2 425 17 440)(3 424 18 439)(4 423 19 438)(5 422 20 437)(6 421 21 436)(7 450 22 435)(8 449 23 434)(9 448 24 433)(10 447 25 432)(11 446 26 431)(12 445 27 430)(13 444 28 429)(14 443 29 428)(15 442 30 427)(31 469 46 454)(32 468 47 453)(33 467 48 452)(34 466 49 451)(35 465 50 480)(36 464 51 479)(37 463 52 478)(38 462 53 477)(39 461 54 476)(40 460 55 475)(41 459 56 474)(42 458 57 473)(43 457 58 472)(44 456 59 471)(45 455 60 470)(61 387 76 372)(62 386 77 371)(63 385 78 370)(64 384 79 369)(65 383 80 368)(66 382 81 367)(67 381 82 366)(68 380 83 365)(69 379 84 364)(70 378 85 363)(71 377 86 362)(72 376 87 361)(73 375 88 390)(74 374 89 389)(75 373 90 388)(91 416 106 401)(92 415 107 400)(93 414 108 399)(94 413 109 398)(95 412 110 397)(96 411 111 396)(97 410 112 395)(98 409 113 394)(99 408 114 393)(100 407 115 392)(101 406 116 391)(102 405 117 420)(103 404 118 419)(104 403 119 418)(105 402 120 417)(121 311 136 326)(122 310 137 325)(123 309 138 324)(124 308 139 323)(125 307 140 322)(126 306 141 321)(127 305 142 320)(128 304 143 319)(129 303 144 318)(130 302 145 317)(131 301 146 316)(132 330 147 315)(133 329 148 314)(134 328 149 313)(135 327 150 312)(151 339 166 354)(152 338 167 353)(153 337 168 352)(154 336 169 351)(155 335 170 350)(156 334 171 349)(157 333 172 348)(158 332 173 347)(159 331 174 346)(160 360 175 345)(161 359 176 344)(162 358 177 343)(163 357 178 342)(164 356 179 341)(165 355 180 340)(181 242 196 257)(182 241 197 256)(183 270 198 255)(184 269 199 254)(185 268 200 253)(186 267 201 252)(187 266 202 251)(188 265 203 250)(189 264 204 249)(190 263 205 248)(191 262 206 247)(192 261 207 246)(193 260 208 245)(194 259 209 244)(195 258 210 243)(211 291 226 276)(212 290 227 275)(213 289 228 274)(214 288 229 273)(215 287 230 272)(216 286 231 271)(217 285 232 300)(218 284 233 299)(219 283 234 298)(220 282 235 297)(221 281 236 296)(222 280 237 295)(223 279 238 294)(224 278 239 293)(225 277 240 292)
G:=sub<Sym(480)| (1,121)(2,122)(3,123)(4,124)(5,125)(6,126)(7,127)(8,128)(9,129)(10,130)(11,131)(12,132)(13,133)(14,134)(15,135)(16,136)(17,137)(18,138)(19,139)(20,140)(21,141)(22,142)(23,143)(24,144)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,166)(32,167)(33,168)(34,169)(35,170)(36,171)(37,172)(38,173)(39,174)(40,175)(41,176)(42,177)(43,178)(44,179)(45,180)(46,151)(47,152)(48,153)(49,154)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,161)(57,162)(58,163)(59,164)(60,165)(61,196)(62,197)(63,198)(64,199)(65,200)(66,201)(67,202)(68,203)(69,204)(70,205)(71,206)(72,207)(73,208)(74,209)(75,210)(76,181)(77,182)(78,183)(79,184)(80,185)(81,186)(82,187)(83,188)(84,189)(85,190)(86,191)(87,192)(88,193)(89,194)(90,195)(91,226)(92,227)(93,228)(94,229)(95,230)(96,231)(97,232)(98,233)(99,234)(100,235)(101,236)(102,237)(103,238)(104,239)(105,240)(106,211)(107,212)(108,213)(109,214)(110,215)(111,216)(112,217)(113,218)(114,219)(115,220)(116,221)(117,222)(118,223)(119,224)(120,225)(241,371)(242,372)(243,373)(244,374)(245,375)(246,376)(247,377)(248,378)(249,379)(250,380)(251,381)(252,382)(253,383)(254,384)(255,385)(256,386)(257,387)(258,388)(259,389)(260,390)(261,361)(262,362)(263,363)(264,364)(265,365)(266,366)(267,367)(268,368)(269,369)(270,370)(271,411)(272,412)(273,413)(274,414)(275,415)(276,416)(277,417)(278,418)(279,419)(280,420)(281,391)(282,392)(283,393)(284,394)(285,395)(286,396)(287,397)(288,398)(289,399)(290,400)(291,401)(292,402)(293,403)(294,404)(295,405)(296,406)(297,407)(298,408)(299,409)(300,410)(301,446)(302,447)(303,448)(304,449)(305,450)(306,421)(307,422)(308,423)(309,424)(310,425)(311,426)(312,427)(313,428)(314,429)(315,430)(316,431)(317,432)(318,433)(319,434)(320,435)(321,436)(322,437)(323,438)(324,439)(325,440)(326,441)(327,442)(328,443)(329,444)(330,445)(331,476)(332,477)(333,478)(334,479)(335,480)(336,451)(337,452)(338,453)(339,454)(340,455)(341,456)(342,457)(343,458)(344,459)(345,460)(346,461)(347,462)(348,463)(349,464)(350,465)(351,466)(352,467)(353,468)(354,469)(355,470)(356,471)(357,472)(358,473)(359,474)(360,475), (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,113)(9,114)(10,115)(11,116)(12,117)(13,118)(14,119)(15,120)(16,91)(17,92)(18,93)(19,94)(20,95)(21,96)(22,97)(23,98)(24,99)(25,100)(26,101)(27,102)(28,103)(29,104)(30,105)(31,69)(32,70)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,81)(44,82)(45,83)(46,84)(47,85)(48,86)(49,87)(50,88)(51,89)(52,90)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68)(121,211)(122,212)(123,213)(124,214)(125,215)(126,216)(127,217)(128,218)(129,219)(130,220)(131,221)(132,222)(133,223)(134,224)(135,225)(136,226)(137,227)(138,228)(139,229)(140,230)(141,231)(142,232)(143,233)(144,234)(145,235)(146,236)(147,237)(148,238)(149,239)(150,240)(151,189)(152,190)(153,191)(154,192)(155,193)(156,194)(157,195)(158,196)(159,197)(160,198)(161,199)(162,200)(163,201)(164,202)(165,203)(166,204)(167,205)(168,206)(169,207)(170,208)(171,209)(172,210)(173,181)(174,182)(175,183)(176,184)(177,185)(178,186)(179,187)(180,188)(241,346)(242,347)(243,348)(244,349)(245,350)(246,351)(247,352)(248,353)(249,354)(250,355)(251,356)(252,357)(253,358)(254,359)(255,360)(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)(271,321)(272,322)(273,323)(274,324)(275,325)(276,326)(277,327)(278,328)(279,329)(280,330)(281,301)(282,302)(283,303)(284,304)(285,305)(286,306)(287,307)(288,308)(289,309)(290,310)(291,311)(292,312)(293,313)(294,314)(295,315)(296,316)(297,317)(298,318)(299,319)(300,320)(361,451)(362,452)(363,453)(364,454)(365,455)(366,456)(367,457)(368,458)(369,459)(370,460)(371,461)(372,462)(373,463)(374,464)(375,465)(376,466)(377,467)(378,468)(379,469)(380,470)(381,471)(382,472)(383,473)(384,474)(385,475)(386,476)(387,477)(388,478)(389,479)(390,480)(391,446)(392,447)(393,448)(394,449)(395,450)(396,421)(397,422)(398,423)(399,424)(400,425)(401,426)(402,427)(403,428)(404,429)(405,430)(406,431)(407,432)(408,433)(409,434)(410,435)(411,436)(412,437)(413,438)(414,439)(415,440)(416,441)(417,442)(418,443)(419,444)(420,445), (1,59)(2,60)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(61,100)(62,101)(63,102)(64,103)(65,104)(66,105)(67,106)(68,107)(69,108)(70,109)(71,110)(72,111)(73,112)(74,113)(75,114)(76,115)(77,116)(78,117)(79,118)(80,119)(81,120)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,97)(89,98)(90,99)(121,164)(122,165)(123,166)(124,167)(125,168)(126,169)(127,170)(128,171)(129,172)(130,173)(131,174)(132,175)(133,176)(134,177)(135,178)(136,179)(137,180)(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)(181,220)(182,221)(183,222)(184,223)(185,224)(186,225)(187,226)(188,227)(189,228)(190,229)(191,230)(192,231)(193,232)(194,233)(195,234)(196,235)(197,236)(198,237)(199,238)(200,239)(201,240)(202,211)(203,212)(204,213)(205,214)(206,215)(207,216)(208,217)(209,218)(210,219)(241,281)(242,282)(243,283)(244,284)(245,285)(246,286)(247,287)(248,288)(249,289)(250,290)(251,291)(252,292)(253,293)(254,294)(255,295)(256,296)(257,297)(258,298)(259,299)(260,300)(261,271)(262,272)(263,273)(264,274)(265,275)(266,276)(267,277)(268,278)(269,279)(270,280)(301,346)(302,347)(303,348)(304,349)(305,350)(306,351)(307,352)(308,353)(309,354)(310,355)(311,356)(312,357)(313,358)(314,359)(315,360)(316,331)(317,332)(318,333)(319,334)(320,335)(321,336)(322,337)(323,338)(324,339)(325,340)(326,341)(327,342)(328,343)(329,344)(330,345)(361,411)(362,412)(363,413)(364,414)(365,415)(366,416)(367,417)(368,418)(369,419)(370,420)(371,391)(372,392)(373,393)(374,394)(375,395)(376,396)(377,397)(378,398)(379,399)(380,400)(381,401)(382,402)(383,403)(384,404)(385,405)(386,406)(387,407)(388,408)(389,409)(390,410)(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)(436,451)(437,452)(438,453)(439,454)(440,455)(441,456)(442,457)(443,458)(444,459)(445,460)(446,461)(447,462)(448,463)(449,464)(450,465), (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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(1,426,16,441)(2,425,17,440)(3,424,18,439)(4,423,19,438)(5,422,20,437)(6,421,21,436)(7,450,22,435)(8,449,23,434)(9,448,24,433)(10,447,25,432)(11,446,26,431)(12,445,27,430)(13,444,28,429)(14,443,29,428)(15,442,30,427)(31,469,46,454)(32,468,47,453)(33,467,48,452)(34,466,49,451)(35,465,50,480)(36,464,51,479)(37,463,52,478)(38,462,53,477)(39,461,54,476)(40,460,55,475)(41,459,56,474)(42,458,57,473)(43,457,58,472)(44,456,59,471)(45,455,60,470)(61,387,76,372)(62,386,77,371)(63,385,78,370)(64,384,79,369)(65,383,80,368)(66,382,81,367)(67,381,82,366)(68,380,83,365)(69,379,84,364)(70,378,85,363)(71,377,86,362)(72,376,87,361)(73,375,88,390)(74,374,89,389)(75,373,90,388)(91,416,106,401)(92,415,107,400)(93,414,108,399)(94,413,109,398)(95,412,110,397)(96,411,111,396)(97,410,112,395)(98,409,113,394)(99,408,114,393)(100,407,115,392)(101,406,116,391)(102,405,117,420)(103,404,118,419)(104,403,119,418)(105,402,120,417)(121,311,136,326)(122,310,137,325)(123,309,138,324)(124,308,139,323)(125,307,140,322)(126,306,141,321)(127,305,142,320)(128,304,143,319)(129,303,144,318)(130,302,145,317)(131,301,146,316)(132,330,147,315)(133,329,148,314)(134,328,149,313)(135,327,150,312)(151,339,166,354)(152,338,167,353)(153,337,168,352)(154,336,169,351)(155,335,170,350)(156,334,171,349)(157,333,172,348)(158,332,173,347)(159,331,174,346)(160,360,175,345)(161,359,176,344)(162,358,177,343)(163,357,178,342)(164,356,179,341)(165,355,180,340)(181,242,196,257)(182,241,197,256)(183,270,198,255)(184,269,199,254)(185,268,200,253)(186,267,201,252)(187,266,202,251)(188,265,203,250)(189,264,204,249)(190,263,205,248)(191,262,206,247)(192,261,207,246)(193,260,208,245)(194,259,209,244)(195,258,210,243)(211,291,226,276)(212,290,227,275)(213,289,228,274)(214,288,229,273)(215,287,230,272)(216,286,231,271)(217,285,232,300)(218,284,233,299)(219,283,234,298)(220,282,235,297)(221,281,236,296)(222,280,237,295)(223,279,238,294)(224,278,239,293)(225,277,240,292) );
G=PermutationGroup([[(1,121),(2,122),(3,123),(4,124),(5,125),(6,126),(7,127),(8,128),(9,129),(10,130),(11,131),(12,132),(13,133),(14,134),(15,135),(16,136),(17,137),(18,138),(19,139),(20,140),(21,141),(22,142),(23,143),(24,144),(25,145),(26,146),(27,147),(28,148),(29,149),(30,150),(31,166),(32,167),(33,168),(34,169),(35,170),(36,171),(37,172),(38,173),(39,174),(40,175),(41,176),(42,177),(43,178),(44,179),(45,180),(46,151),(47,152),(48,153),(49,154),(50,155),(51,156),(52,157),(53,158),(54,159),(55,160),(56,161),(57,162),(58,163),(59,164),(60,165),(61,196),(62,197),(63,198),(64,199),(65,200),(66,201),(67,202),(68,203),(69,204),(70,205),(71,206),(72,207),(73,208),(74,209),(75,210),(76,181),(77,182),(78,183),(79,184),(80,185),(81,186),(82,187),(83,188),(84,189),(85,190),(86,191),(87,192),(88,193),(89,194),(90,195),(91,226),(92,227),(93,228),(94,229),(95,230),(96,231),(97,232),(98,233),(99,234),(100,235),(101,236),(102,237),(103,238),(104,239),(105,240),(106,211),(107,212),(108,213),(109,214),(110,215),(111,216),(112,217),(113,218),(114,219),(115,220),(116,221),(117,222),(118,223),(119,224),(120,225),(241,371),(242,372),(243,373),(244,374),(245,375),(246,376),(247,377),(248,378),(249,379),(250,380),(251,381),(252,382),(253,383),(254,384),(255,385),(256,386),(257,387),(258,388),(259,389),(260,390),(261,361),(262,362),(263,363),(264,364),(265,365),(266,366),(267,367),(268,368),(269,369),(270,370),(271,411),(272,412),(273,413),(274,414),(275,415),(276,416),(277,417),(278,418),(279,419),(280,420),(281,391),(282,392),(283,393),(284,394),(285,395),(286,396),(287,397),(288,398),(289,399),(290,400),(291,401),(292,402),(293,403),(294,404),(295,405),(296,406),(297,407),(298,408),(299,409),(300,410),(301,446),(302,447),(303,448),(304,449),(305,450),(306,421),(307,422),(308,423),(309,424),(310,425),(311,426),(312,427),(313,428),(314,429),(315,430),(316,431),(317,432),(318,433),(319,434),(320,435),(321,436),(322,437),(323,438),(324,439),(325,440),(326,441),(327,442),(328,443),(329,444),(330,445),(331,476),(332,477),(333,478),(334,479),(335,480),(336,451),(337,452),(338,453),(339,454),(340,455),(341,456),(342,457),(343,458),(344,459),(345,460),(346,461),(347,462),(348,463),(349,464),(350,465),(351,466),(352,467),(353,468),(354,469),(355,470),(356,471),(357,472),(358,473),(359,474),(360,475)], [(1,106),(2,107),(3,108),(4,109),(5,110),(6,111),(7,112),(8,113),(9,114),(10,115),(11,116),(12,117),(13,118),(14,119),(15,120),(16,91),(17,92),(18,93),(19,94),(20,95),(21,96),(22,97),(23,98),(24,99),(25,100),(26,101),(27,102),(28,103),(29,104),(30,105),(31,69),(32,70),(33,71),(34,72),(35,73),(36,74),(37,75),(38,76),(39,77),(40,78),(41,79),(42,80),(43,81),(44,82),(45,83),(46,84),(47,85),(48,86),(49,87),(50,88),(51,89),(52,90),(53,61),(54,62),(55,63),(56,64),(57,65),(58,66),(59,67),(60,68),(121,211),(122,212),(123,213),(124,214),(125,215),(126,216),(127,217),(128,218),(129,219),(130,220),(131,221),(132,222),(133,223),(134,224),(135,225),(136,226),(137,227),(138,228),(139,229),(140,230),(141,231),(142,232),(143,233),(144,234),(145,235),(146,236),(147,237),(148,238),(149,239),(150,240),(151,189),(152,190),(153,191),(154,192),(155,193),(156,194),(157,195),(158,196),(159,197),(160,198),(161,199),(162,200),(163,201),(164,202),(165,203),(166,204),(167,205),(168,206),(169,207),(170,208),(171,209),(172,210),(173,181),(174,182),(175,183),(176,184),(177,185),(178,186),(179,187),(180,188),(241,346),(242,347),(243,348),(244,349),(245,350),(246,351),(247,352),(248,353),(249,354),(250,355),(251,356),(252,357),(253,358),(254,359),(255,360),(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),(271,321),(272,322),(273,323),(274,324),(275,325),(276,326),(277,327),(278,328),(279,329),(280,330),(281,301),(282,302),(283,303),(284,304),(285,305),(286,306),(287,307),(288,308),(289,309),(290,310),(291,311),(292,312),(293,313),(294,314),(295,315),(296,316),(297,317),(298,318),(299,319),(300,320),(361,451),(362,452),(363,453),(364,454),(365,455),(366,456),(367,457),(368,458),(369,459),(370,460),(371,461),(372,462),(373,463),(374,464),(375,465),(376,466),(377,467),(378,468),(379,469),(380,470),(381,471),(382,472),(383,473),(384,474),(385,475),(386,476),(387,477),(388,478),(389,479),(390,480),(391,446),(392,447),(393,448),(394,449),(395,450),(396,421),(397,422),(398,423),(399,424),(400,425),(401,426),(402,427),(403,428),(404,429),(405,430),(406,431),(407,432),(408,433),(409,434),(410,435),(411,436),(412,437),(413,438),(414,439),(415,440),(416,441),(417,442),(418,443),(419,444),(420,445)], [(1,59),(2,60),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(61,100),(62,101),(63,102),(64,103),(65,104),(66,105),(67,106),(68,107),(69,108),(70,109),(71,110),(72,111),(73,112),(74,113),(75,114),(76,115),(77,116),(78,117),(79,118),(80,119),(81,120),(82,91),(83,92),(84,93),(85,94),(86,95),(87,96),(88,97),(89,98),(90,99),(121,164),(122,165),(123,166),(124,167),(125,168),(126,169),(127,170),(128,171),(129,172),(130,173),(131,174),(132,175),(133,176),(134,177),(135,178),(136,179),(137,180),(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),(181,220),(182,221),(183,222),(184,223),(185,224),(186,225),(187,226),(188,227),(189,228),(190,229),(191,230),(192,231),(193,232),(194,233),(195,234),(196,235),(197,236),(198,237),(199,238),(200,239),(201,240),(202,211),(203,212),(204,213),(205,214),(206,215),(207,216),(208,217),(209,218),(210,219),(241,281),(242,282),(243,283),(244,284),(245,285),(246,286),(247,287),(248,288),(249,289),(250,290),(251,291),(252,292),(253,293),(254,294),(255,295),(256,296),(257,297),(258,298),(259,299),(260,300),(261,271),(262,272),(263,273),(264,274),(265,275),(266,276),(267,277),(268,278),(269,279),(270,280),(301,346),(302,347),(303,348),(304,349),(305,350),(306,351),(307,352),(308,353),(309,354),(310,355),(311,356),(312,357),(313,358),(314,359),(315,360),(316,331),(317,332),(318,333),(319,334),(320,335),(321,336),(322,337),(323,338),(324,339),(325,340),(326,341),(327,342),(328,343),(329,344),(330,345),(361,411),(362,412),(363,413),(364,414),(365,415),(366,416),(367,417),(368,418),(369,419),(370,420),(371,391),(372,392),(373,393),(374,394),(375,395),(376,396),(377,397),(378,398),(379,399),(380,400),(381,401),(382,402),(383,403),(384,404),(385,405),(386,406),(387,407),(388,408),(389,409),(390,410),(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),(436,451),(437,452),(438,453),(439,454),(440,455),(441,456),(442,457),(443,458),(444,459),(445,460),(446,461),(447,462),(448,463),(449,464),(450,465)], [(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)], [(1,426,16,441),(2,425,17,440),(3,424,18,439),(4,423,19,438),(5,422,20,437),(6,421,21,436),(7,450,22,435),(8,449,23,434),(9,448,24,433),(10,447,25,432),(11,446,26,431),(12,445,27,430),(13,444,28,429),(14,443,29,428),(15,442,30,427),(31,469,46,454),(32,468,47,453),(33,467,48,452),(34,466,49,451),(35,465,50,480),(36,464,51,479),(37,463,52,478),(38,462,53,477),(39,461,54,476),(40,460,55,475),(41,459,56,474),(42,458,57,473),(43,457,58,472),(44,456,59,471),(45,455,60,470),(61,387,76,372),(62,386,77,371),(63,385,78,370),(64,384,79,369),(65,383,80,368),(66,382,81,367),(67,381,82,366),(68,380,83,365),(69,379,84,364),(70,378,85,363),(71,377,86,362),(72,376,87,361),(73,375,88,390),(74,374,89,389),(75,373,90,388),(91,416,106,401),(92,415,107,400),(93,414,108,399),(94,413,109,398),(95,412,110,397),(96,411,111,396),(97,410,112,395),(98,409,113,394),(99,408,114,393),(100,407,115,392),(101,406,116,391),(102,405,117,420),(103,404,118,419),(104,403,119,418),(105,402,120,417),(121,311,136,326),(122,310,137,325),(123,309,138,324),(124,308,139,323),(125,307,140,322),(126,306,141,321),(127,305,142,320),(128,304,143,319),(129,303,144,318),(130,302,145,317),(131,301,146,316),(132,330,147,315),(133,329,148,314),(134,328,149,313),(135,327,150,312),(151,339,166,354),(152,338,167,353),(153,337,168,352),(154,336,169,351),(155,335,170,350),(156,334,171,349),(157,333,172,348),(158,332,173,347),(159,331,174,346),(160,360,175,345),(161,359,176,344),(162,358,177,343),(163,357,178,342),(164,356,179,341),(165,355,180,340),(181,242,196,257),(182,241,197,256),(183,270,198,255),(184,269,199,254),(185,268,200,253),(186,267,201,252),(187,266,202,251),(188,265,203,250),(189,264,204,249),(190,263,205,248),(191,262,206,247),(192,261,207,246),(193,260,208,245),(194,259,209,244),(195,258,210,243),(211,291,226,276),(212,290,227,275),(213,289,228,274),(214,288,229,273),(215,287,230,272),(216,286,231,271),(217,285,232,300),(218,284,233,299),(219,283,234,298),(220,282,235,297),(221,281,236,296),(222,280,237,295),(223,279,238,294),(224,278,239,293),(225,277,240,292)]])
144 conjugacy classes
| class | 1 | 2A | ··· | 2O | 3 | 4A | ··· | 4P | 5A | 5B | 6A | ··· | 6O | 10A | ··· | 10AD | 15A | 15B | 15C | 15D | 30A | ··· | 30BH |
| order | 1 | 2 | ··· | 2 | 3 | 4 | ··· | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 30 | ··· | 30 |
| size | 1 | 1 | ··· | 1 | 2 | 15 | ··· | 15 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | S3 | D5 | Dic3 | D6 | Dic5 | D10 | D15 | Dic15 | D30 |
| kernel | C23×Dic15 | C22×Dic15 | C23×C30 | C22×C30 | C23×C10 | C23×C6 | C22×C10 | C22×C10 | C22×C6 | C22×C6 | C24 | C23 | C23 |
| # reps | 1 | 14 | 1 | 16 | 1 | 2 | 8 | 7 | 16 | 14 | 4 | 32 | 28 |
Matrix representation of C23×Dic15 ►in GL7(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 60 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 50 | 49 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 49 | 55 | 0 | 0 | 0 | 0 |
| 0 | 34 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 55 | 47 | 0 | 0 |
| 0 | 0 | 0 | 7 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 13 |
| 0 | 0 | 0 | 0 | 0 | 17 | 49 |
G:=sub<GL(7,GF(61))| [1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,1,17,0,0,0,0,0,0,0,27,2,0,0,0,0,0,0,52,0,0,0,0,0,0,0,5,50,0,0,0,0,0,0,49],[60,0,0,0,0,0,0,0,49,34,0,0,0,0,0,55,12,0,0,0,0,0,0,0,55,7,0,0,0,0,0,47,6,0,0,0,0,0,0,0,12,17,0,0,0,0,0,13,49] >;
C23×Dic15 in GAP, Magma, Sage, TeX
C_2^3\times {\rm Dic}_{15} % in TeX
G:=Group("C2^3xDic15"); // GroupNames label
G:=SmallGroup(480,1178);
// by ID
G=gap.SmallGroup(480,1178);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^30=1,e^2=d^15,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,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations