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G = C22×Dic30  order 480 = 25·3·5

Direct product of C22 and Dic30

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

Aliases: C22×Dic30, C30.54C24, C23.38D30, C60.250C23, Dic15.26C23, C30⋊5(C2×Q8), (C2×C30)⋊10Q8, C15⋊6(C22×Q8), C6⋊3(C2×Dic10), C10⋊3(C2×Dic6), (C2×C6)⋊9Dic10, (C2×C4).87D30, C5⋊3(C22×Dic6), (C2×C10)⋊12Dic6, (C2×C20).396D6, (C22×C12).8D5, C6.54(C23×D5), C2.3(C23×D15), C3⋊3(C22×Dic10), (C2×C12).401D10, (C22×C20).12S3, (C22×C60).11C2, C10.54(S3×C23), C4.31(C22×D15), (C22×C4).10D15, (C2×C30).318C23, C20.221(C22×S3), (C2×C60).482C22, (C22×C10).142D6, C12.223(C22×D5), (C22×C6).124D10, (C22×Dic15).6C2, C22.28(C22×D15), (C22×C30).147C22, (C2×Dic15).176C22, (C2×C6).314(C22×D5), (C2×C10).313(C22×S3), SmallGroup(480,1165)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C30 — C22×Dic30
Chief series
C1 — C5 — C15 — C30 — Dic15 — C2×Dic15 — C22×Dic15 — C22×Dic30
Lower central
C15 — C30 — C22×Dic30

Subgroups: 1332 in 312 conjugacy classes, 159 normal (17 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C4 [×8], C22 [×7], C5, C6, C6 [×6], C2×C4 [×6], C2×C4 [×12], Q8 [×16], C23, C10, C10 [×6], Dic3 [×8], C12 [×4], C2×C6 [×7], C15, C22×C4, C22×C4 [×2], C2×Q8 [×12], Dic5 [×8], C20 [×4], C2×C10 [×7], Dic6 [×16], C2×Dic3 [×12], C2×C12 [×6], C22×C6, C30, C30 [×6], C22×Q8, Dic10 [×16], C2×Dic5 [×12], C2×C20 [×6], C22×C10, C2×Dic6 [×12], C22×Dic3 [×2], C22×C12, Dic15 [×8], C60 [×4], C2×C30 [×7], C2×Dic10 [×12], C22×Dic5 [×2], C22×C20, C22×Dic6, Dic30 [×16], C2×Dic15 [×12], C2×C60 [×6], C22×C30, C22×Dic10, C2×Dic30 [×12], C22×Dic15 [×2], C22×C60, C22×Dic30

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], D15, C22×Q8, Dic10 [×4], C22×D5 [×7], C2×Dic6 [×6], S3×C23, D30 [×7], C2×Dic10 [×6], C23×D5, C22×Dic6, Dic30 [×4], C22×D15 [×7], C22×Dic10, C2×Dic30 [×6], C23×D15, C22×Dic30

Generators and relations
 G = < a,b,c,d | a2=b2=c60=1, d2=c30, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

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

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

(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)

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

Matrix representation ►G ⊆ GL6(𝔽61)

100000
010000
001000
000100
0000600
0000060
,
100000
010000
0060000
0006000
000010
000001
,
15380000
23380000
00462300
00382300
0000601
00001644
,
8490000
41530000
0084900
00415300
00003138
00005530

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,23,0,0,0,0,38,38,0,0,0,0,0,0,46,38,0,0,0,0,23,23,0,0,0,0,0,0,60,16,0,0,0,0,1,44],[8,41,0,0,0,0,49,53,0,0,0,0,0,0,8,41,0,0,0,0,49,53,0,0,0,0,0,0,31,55,0,0,0,0,38,30] >;
 

132 conjugacy classes

class 1 2A···2G 3 4A4B4C4D4E···4L5A5B6A···6G10A···10N12A···12H15A15B15C15D20A···20P30A···30AB60A···60AF
order12···2344444···4556···610···1012···121515151520···2030···3060···60
size11···12222230···30222···22···22···222222···22···22···2

132 irreducible representations

dim11112222222222222
type+++++-+++++-+-++-
imageC1C2C2C2S3Q8D5D6D6D10D10Dic6D15Dic10D30D30Dic30
kernelC22×Dic30C2×Dic30C22×Dic15C22×C60C22×C20C2×C30C22×C12C2×C20C22×C10C2×C12C22×C6C2×C10C22×C4C2×C6C2×C4C23C22
# reps1122114261122841624432

In GAP, Magma, Sage, TeX 

C_2^2\times Dic_{30}
 
% in TeX
 
G:=Group("C2^2xDic30");
 
// GroupNames label
 
G:=SmallGroup(480,1165);
 
// by ID
 
G=gap.SmallGroup(480,1165);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,675,80,2693,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^60=1,d^2=c^30,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

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