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G = C22×Dic30order 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, C305(C2×Q8), (C2×C30)⋊10Q8, C156(C22×Q8), (C2×C6)⋊9Dic10, C103(C2×Dic6), C63(C2×Dic10), (C2×C4).87D30, C53(C22×Dic6), (C2×C10)⋊12Dic6, (C2×C20).396D6, C6.54(C23×D5), (C22×C12).8D5, C2.3(C23×D15), C33(C22×Dic10), (C2×C12).401D10, (C22×C20).12S3, (C22×C60).11C2, C10.54(S3×C23), C4.31(C22×D15), (C22×C4).10D15, C20.221(C22×S3), (C2×C30).318C23, (C2×C60).482C22, C12.223(C22×D5), (C22×C10).142D6, (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

C1C30 — C22×Dic30
C1C5C15C30Dic15C2×Dic15C22×Dic15 — C22×Dic30
C15C30 — C22×Dic30
C1C23C22×C4

Generators and relations for C22×Dic30
 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 >

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

Smallest permutation representation of C22×Dic30
Regular action on 480 points
Generators in S480
(1 355)(2 356)(3 357)(4 358)(5 359)(6 360)(7 301)(8 302)(9 303)(10 304)(11 305)(12 306)(13 307)(14 308)(15 309)(16 310)(17 311)(18 312)(19 313)(20 314)(21 315)(22 316)(23 317)(24 318)(25 319)(26 320)(27 321)(28 322)(29 323)(30 324)(31 325)(32 326)(33 327)(34 328)(35 329)(36 330)(37 331)(38 332)(39 333)(40 334)(41 335)(42 336)(43 337)(44 338)(45 339)(46 340)(47 341)(48 342)(49 343)(50 344)(51 345)(52 346)(53 347)(54 348)(55 349)(56 350)(57 351)(58 352)(59 353)(60 354)(61 430)(62 431)(63 432)(64 433)(65 434)(66 435)(67 436)(68 437)(69 438)(70 439)(71 440)(72 441)(73 442)(74 443)(75 444)(76 445)(77 446)(78 447)(79 448)(80 449)(81 450)(82 451)(83 452)(84 453)(85 454)(86 455)(87 456)(88 457)(89 458)(90 459)(91 460)(92 461)(93 462)(94 463)(95 464)(96 465)(97 466)(98 467)(99 468)(100 469)(101 470)(102 471)(103 472)(104 473)(105 474)(106 475)(107 476)(108 477)(109 478)(110 479)(111 480)(112 421)(113 422)(114 423)(115 424)(116 425)(117 426)(118 427)(119 428)(120 429)(121 277)(122 278)(123 279)(124 280)(125 281)(126 282)(127 283)(128 284)(129 285)(130 286)(131 287)(132 288)(133 289)(134 290)(135 291)(136 292)(137 293)(138 294)(139 295)(140 296)(141 297)(142 298)(143 299)(144 300)(145 241)(146 242)(147 243)(148 244)(149 245)(150 246)(151 247)(152 248)(153 249)(154 250)(155 251)(156 252)(157 253)(158 254)(159 255)(160 256)(161 257)(162 258)(163 259)(164 260)(165 261)(166 262)(167 263)(168 264)(169 265)(170 266)(171 267)(172 268)(173 269)(174 270)(175 271)(176 272)(177 273)(178 274)(179 275)(180 276)(181 385)(182 386)(183 387)(184 388)(185 389)(186 390)(187 391)(188 392)(189 393)(190 394)(191 395)(192 396)(193 397)(194 398)(195 399)(196 400)(197 401)(198 402)(199 403)(200 404)(201 405)(202 406)(203 407)(204 408)(205 409)(206 410)(207 411)(208 412)(209 413)(210 414)(211 415)(212 416)(213 417)(214 418)(215 419)(216 420)(217 361)(218 362)(219 363)(220 364)(221 365)(222 366)(223 367)(224 368)(225 369)(226 370)(227 371)(228 372)(229 373)(230 374)(231 375)(232 376)(233 377)(234 378)(235 379)(236 380)(237 381)(238 382)(239 383)(240 384)
(1 122)(2 123)(3 124)(4 125)(5 126)(6 127)(7 128)(8 129)(9 130)(10 131)(11 132)(12 133)(13 134)(14 135)(15 136)(16 137)(17 138)(18 139)(19 140)(20 141)(21 142)(22 143)(23 144)(24 145)(25 146)(26 147)(27 148)(28 149)(29 150)(30 151)(31 152)(32 153)(33 154)(34 155)(35 156)(36 157)(37 158)(38 159)(39 160)(40 161)(41 162)(42 163)(43 164)(44 165)(45 166)(46 167)(47 168)(48 169)(49 170)(50 171)(51 172)(52 173)(53 174)(54 175)(55 176)(56 177)(57 178)(58 179)(59 180)(60 121)(61 388)(62 389)(63 390)(64 391)(65 392)(66 393)(67 394)(68 395)(69 396)(70 397)(71 398)(72 399)(73 400)(74 401)(75 402)(76 403)(77 404)(78 405)(79 406)(80 407)(81 408)(82 409)(83 410)(84 411)(85 412)(86 413)(87 414)(88 415)(89 416)(90 417)(91 418)(92 419)(93 420)(94 361)(95 362)(96 363)(97 364)(98 365)(99 366)(100 367)(101 368)(102 369)(103 370)(104 371)(105 372)(106 373)(107 374)(108 375)(109 376)(110 377)(111 378)(112 379)(113 380)(114 381)(115 382)(116 383)(117 384)(118 385)(119 386)(120 387)(181 427)(182 428)(183 429)(184 430)(185 431)(186 432)(187 433)(188 434)(189 435)(190 436)(191 437)(192 438)(193 439)(194 440)(195 441)(196 442)(197 443)(198 444)(199 445)(200 446)(201 447)(202 448)(203 449)(204 450)(205 451)(206 452)(207 453)(208 454)(209 455)(210 456)(211 457)(212 458)(213 459)(214 460)(215 461)(216 462)(217 463)(218 464)(219 465)(220 466)(221 467)(222 468)(223 469)(224 470)(225 471)(226 472)(227 473)(228 474)(229 475)(230 476)(231 477)(232 478)(233 479)(234 480)(235 421)(236 422)(237 423)(238 424)(239 425)(240 426)(241 318)(242 319)(243 320)(244 321)(245 322)(246 323)(247 324)(248 325)(249 326)(250 327)(251 328)(252 329)(253 330)(254 331)(255 332)(256 333)(257 334)(258 335)(259 336)(260 337)(261 338)(262 339)(263 340)(264 341)(265 342)(266 343)(267 344)(268 345)(269 346)(270 347)(271 348)(272 349)(273 350)(274 351)(275 352)(276 353)(277 354)(278 355)(279 356)(280 357)(281 358)(282 359)(283 360)(284 301)(285 302)(286 303)(287 304)(288 305)(289 306)(290 307)(291 308)(292 309)(293 310)(294 311)(295 312)(296 313)(297 314)(298 315)(299 316)(300 317)
(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)
(1 361 31 391)(2 420 32 390)(3 419 33 389)(4 418 34 388)(5 417 35 387)(6 416 36 386)(7 415 37 385)(8 414 38 384)(9 413 39 383)(10 412 40 382)(11 411 41 381)(12 410 42 380)(13 409 43 379)(14 408 44 378)(15 407 45 377)(16 406 46 376)(17 405 47 375)(18 404 48 374)(19 403 49 373)(20 402 50 372)(21 401 51 371)(22 400 52 370)(23 399 53 369)(24 398 54 368)(25 397 55 367)(26 396 56 366)(27 395 57 365)(28 394 58 364)(29 393 59 363)(30 392 60 362)(61 125 91 155)(62 124 92 154)(63 123 93 153)(64 122 94 152)(65 121 95 151)(66 180 96 150)(67 179 97 149)(68 178 98 148)(69 177 99 147)(70 176 100 146)(71 175 101 145)(72 174 102 144)(73 173 103 143)(74 172 104 142)(75 171 105 141)(76 170 106 140)(77 169 107 139)(78 168 108 138)(79 167 109 137)(80 166 110 136)(81 165 111 135)(82 164 112 134)(83 163 113 133)(84 162 114 132)(85 161 115 131)(86 160 116 130)(87 159 117 129)(88 158 118 128)(89 157 119 127)(90 156 120 126)(181 301 211 331)(182 360 212 330)(183 359 213 329)(184 358 214 328)(185 357 215 327)(186 356 216 326)(187 355 217 325)(188 354 218 324)(189 353 219 323)(190 352 220 322)(191 351 221 321)(192 350 222 320)(193 349 223 319)(194 348 224 318)(195 347 225 317)(196 346 226 316)(197 345 227 315)(198 344 228 314)(199 343 229 313)(200 342 230 312)(201 341 231 311)(202 340 232 310)(203 339 233 309)(204 338 234 308)(205 337 235 307)(206 336 236 306)(207 335 237 305)(208 334 238 304)(209 333 239 303)(210 332 240 302)(241 440 271 470)(242 439 272 469)(243 438 273 468)(244 437 274 467)(245 436 275 466)(246 435 276 465)(247 434 277 464)(248 433 278 463)(249 432 279 462)(250 431 280 461)(251 430 281 460)(252 429 282 459)(253 428 283 458)(254 427 284 457)(255 426 285 456)(256 425 286 455)(257 424 287 454)(258 423 288 453)(259 422 289 452)(260 421 290 451)(261 480 291 450)(262 479 292 449)(263 478 293 448)(264 477 294 447)(265 476 295 446)(266 475 296 445)(267 474 297 444)(268 473 298 443)(269 472 299 442)(270 471 300 441)

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

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

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

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

Matrix representation of C22×Dic30 in 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] >;

C22×Dic30 in GAP, Magma, Sage, TeX

C_2^2\times {\rm 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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