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## G = C22×Dic30order 480 = 25·3·5

### Direct product of C22 and Dic30

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
 Upper central C1 — C23 — C22×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, C3, C4, C4, C22, C5, C6, C6, C2×C4, C2×C4, Q8, C23, C10, C10, Dic3, C12, C2×C6, C15, C22×C4, C22×C4, C2×Q8, Dic5, C20, C2×C10, Dic6, C2×Dic3, C2×C12, C22×C6, C30, C30, C22×Q8, Dic10, C2×Dic5, C2×C20, C22×C10, C2×Dic6, C22×Dic3, C22×C12, Dic15, C60, C2×C30, C2×Dic10, C22×Dic5, C22×C20, C22×Dic6, Dic30, C2×Dic15, C2×C60, C22×C30, C22×Dic10, C2×Dic30, C22×Dic15, C22×C60, C22×Dic30
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, C24, D10, Dic6, C22×S3, D15, C22×Q8, Dic10, C22×D5, C2×Dic6, S3×C23, D30, C2×Dic10, C23×D5, C22×Dic6, Dic30, C22×D15, C22×Dic10, C2×Dic30, C23×D15, C22×Dic30

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

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

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

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

132 conjugacy classes

 class 1 2A ··· 2G 3 4A 4B 4C 4D 4E ··· 4L 5A 5B 6A ··· 6G 10A ··· 10N 12A ··· 12H 15A 15B 15C 15D 20A ··· 20P 30A ··· 30AB 60A ··· 60AF order 1 2 ··· 2 3 4 4 4 4 4 ··· 4 5 5 6 ··· 6 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 ··· 1 2 2 2 2 2 30 ··· 30 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

132 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - + + + + + - + - + + - image C1 C2 C2 C2 S3 Q8 D5 D6 D6 D10 D10 Dic6 D15 Dic10 D30 D30 Dic30 kernel C22×Dic30 C2×Dic30 C22×Dic15 C22×C60 C22×C20 C2×C30 C22×C12 C2×C20 C22×C10 C2×C12 C22×C6 C2×C10 C22×C4 C2×C6 C2×C4 C23 C22 # reps 1 12 2 1 1 4 2 6 1 12 2 8 4 16 24 4 32

Matrix representation of C22×Dic30 in GL6(𝔽61)

 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 38 0 0 0 0 23 38 0 0 0 0 0 0 46 23 0 0 0 0 38 23 0 0 0 0 0 0 60 1 0 0 0 0 16 44
,
 8 49 0 0 0 0 41 53 0 0 0 0 0 0 8 49 0 0 0 0 41 53 0 0 0 0 0 0 31 38 0 0 0 0 55 30

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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