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G = C24×C30  order 480 = 25·3·5

Abelian group of type [2,2,2,2,30]

direct product, abelian, monomial, 2-elementary

Aliases: C24×C30, SmallGroup(480,1213)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C24×C30
Chief series
C1 — C5 — C15 — C30 — C2×C30 — C22×C30 — C23×C30 — C24×C30
Lower central
C1 — C24×C30
Upper central
C1 — C24×C30

Subgroups: 1496, all normal (8 characteristic)
C1, C2 [×31], C3, C22 [×155], C5, C6 [×31], C23 [×155], C10 [×31], C2×C6 [×155], C15, C24 [×31], C2×C10 [×155], C22×C6 [×155], C30 [×31], C25, C22×C10 [×155], C23×C6 [×31], C2×C30 [×155], C23×C10 [×31], C24×C6, C22×C30 [×155], C24×C10, C23×C30 [×31], C24×C30

Quotients:
C1, C2 [×31], C3, C22 [×155], C5, C6 [×31], C23 [×155], C10 [×31], C2×C6 [×155], C15, C24 [×31], C2×C10 [×155], C22×C6 [×155], C30 [×31], C25, C22×C10 [×155], C23×C6 [×31], C2×C30 [×155], C23×C10 [×31], C24×C6, C22×C30 [×155], C24×C10, C23×C30 [×31], C24×C30

Generators and relations
 G = < a,b,c,d,e | a2=b2=c2=d2=e30=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, de=ed >

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

(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)(31 156)(32 157)(33 158)(34 159)(35 160)(36 161)(37 162)(38 163)(39 164)(40 165)(41 166)(42 167)(43 168)(44 169)(45 170)(46 171)(47 172)(48 173)(49 174)(50 175)(51 176)(52 177)(53 178)(54 179)(55 180)(56 151)(57 152)(58 153)(59 154)(60 155)(61 201)(62 202)(63 203)(64 204)(65 205)(66 206)(67 207)(68 208)(69 209)(70 210)(71 181)(72 182)(73 183)(74 184)(75 185)(76 186)(77 187)(78 188)(79 189)(80 190)(81 191)(82 192)(83 193)(84 194)(85 195)(86 196)(87 197)(88 198)(89 199)(90 200)(91 231)(92 232)(93 233)(94 234)(95 235)(96 236)(97 237)(98 238)(99 239)(100 240)(101 211)(102 212)(103 213)(104 214)(105 215)(106 216)(107 217)(108 218)(109 219)(110 220)(111 221)(112 222)(113 223)(114 224)(115 225)(116 226)(117 227)(118 228)(119 229)(120 230)(241 376)(242 377)(243 378)(244 379)(245 380)(246 381)(247 382)(248 383)(249 384)(250 385)(251 386)(252 387)(253 388)(254 389)(255 390)(256 361)(257 362)(258 363)(259 364)(260 365)(261 366)(262 367)(263 368)(264 369)(265 370)(266 371)(267 372)(268 373)(269 374)(270 375)(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)(301 421)(302 422)(303 423)(304 424)(305 425)(306 426)(307 427)(308 428)(309 429)(310 430)(311 431)(312 432)(313 433)(314 434)(315 435)(316 436)(317 437)(318 438)(319 439)(320 440)(321 441)(322 442)(323 443)(324 444)(325 445)(326 446)(327 447)(328 448)(329 449)(330 450)(331 466)(332 467)(333 468)(334 469)(335 470)(336 471)(337 472)(338 473)(339 474)(340 475)(341 476)(342 477)(343 478)(344 479)(345 480)(346 451)(347 452)(348 453)(349 454)(350 455)(351 456)(352 457)(353 458)(354 459)(355 460)(356 461)(357 462)(358 463)(359 464)(360 465)

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

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

Matrix representation ►G ⊆ GL5(𝔽31)

10000
01000
00100
000300
000030
,
10000
01000
003000
000300
000030
,
10000
030000
003000
00010
000030
,
300000
030000
003000
000300
000030
,
300000
011000
001100
000110
00005

G:=sub<GL(5,GF(31))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0,30],[1,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,30],[1,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,30],[30,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,30],[30,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,5] >;
 

480 conjugacy classes

class 1 2A···2AE3A3B5A5B5C5D6A···6BJ10A···10DT15A···15H30A···30IN
order12···23355556···610···1015···1530···30
size11···11111111···11···11···11···1

480 irreducible representations

dim11111111
type++
imageC1C2C3C5C6C10C15C30
kernelC24×C30C23×C30C24×C10C24×C6C23×C10C23×C6C25C24
# reps13124621248248

In GAP, Magma, Sage, TeX 

C_2^4\times C_{30}
 
% in TeX
 
G:=Group("C2^4xC30");
 
// GroupNames label
 
G:=SmallGroup(480,1213);
 
// by ID
 
G=gap.SmallGroup(480,1213);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^2=e^30=1,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,d*e=e*d>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
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