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G = Dic3×C22×C10  order 480 = 25·3·5

Direct product of C22×C10 and Dic3

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

Aliases: Dic3×C22×C10, C30.97C24, C3⋊2(C23×C20), (C22×C6)⋊5C20, C6⋊2(C22×C20), C15⋊13(C23×C4), C24.3(C5×S3), (C22×C30)⋊17C4, C30⋊13(C22×C4), (C23×C10).6S3, (C23×C6).3C10, (C23×C30).7C2, C23.41(S3×C10), C10.82(S3×C23), C6.14(C23×C10), (C2×C30).449C23, (C22×C10).155D6, (C22×C30).185C22, (C2×C6)⋊9(C2×C20), (C2×C30)⋊45(C2×C4), C2.2(S3×C22×C10), C22.33(S3×C2×C10), (C22×C6).47(C2×C10), (C2×C6).69(C22×C10), (C2×C10).380(C22×S3), SmallGroup(480,1163)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — Dic3×C22×C10
Chief series
C1 — C3 — C6 — C30 — C5×Dic3 — C10×Dic3 — Dic3×C2×C10 — Dic3×C22×C10
Lower central
C3 — Dic3×C22×C10

Subgroups: 676 in 472 conjugacy classes, 370 normal (14 characteristic)
C1, C2, C2 [×14], C3, C4 [×8], C22 [×35], C5, C6, C6 [×14], C2×C4 [×28], C23 [×15], C10, C10 [×14], Dic3 [×8], C2×C6 [×35], C15, C22×C4 [×14], C24, C20 [×8], C2×C10 [×35], C2×Dic3 [×28], C22×C6 [×15], C30, C30 [×14], C23×C4, C2×C20 [×28], C22×C10 [×15], C22×Dic3 [×14], C23×C6, C5×Dic3 [×8], C2×C30 [×35], C22×C20 [×14], C23×C10, C23×Dic3, C10×Dic3 [×28], C22×C30 [×15], C23×C20, Dic3×C2×C10 [×14], C23×C30, Dic3×C22×C10

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C5, S3, C2×C4 [×28], C23 [×15], C10 [×15], Dic3 [×8], D6 [×7], C22×C4 [×14], C24, C20 [×8], C2×C10 [×35], C2×Dic3 [×28], C22×S3 [×7], C5×S3, C23×C4, C2×C20 [×28], C22×C10 [×15], C22×Dic3 [×14], S3×C23, C5×Dic3 [×8], S3×C10 [×7], C22×C20 [×14], C23×C10, C23×Dic3, C10×Dic3 [×28], S3×C2×C10 [×7], C23×C20, Dic3×C2×C10 [×14], S3×C22×C10, Dic3×C22×C10

Generators and relations
 G = < a,b,c,d,e | a2=b2=c10=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

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

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

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

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

Matrix representation ►G ⊆ GL5(𝔽61)

10000
01000
006000
000600
000060
,
10000
01000
006000
00010
00001
,
600000
01000
006000
000580
000058
,
10000
060000
00100
000601
000600
,
600000
011000
006000
0002553
0001736

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,58,0,0,0,0,0,58],[1,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,60,60,0,0,0,1,0],[60,0,0,0,0,0,11,0,0,0,0,0,60,0,0,0,0,0,25,17,0,0,0,53,36] >;
 

240 conjugacy classes

class 1 2A···2O 3 4A···4P5A5B5C5D6A···6O10A···10BH15A15B15C15D20A···20BL30A···30BH
order12···234···455556···610···101515151520···2030···30
size11···123···311112···21···122223···32···2

240 irreducible representations

dim11111111222222
type++++-+
imageC1C2C2C4C5C10C10C20S3Dic3D6C5×S3C5×Dic3S3×C10
kernelDic3×C22×C10Dic3×C2×C10C23×C30C22×C30C23×Dic3C22×Dic3C23×C6C22×C6C23×C10C22×C10C22×C10C24C23C23
# reps11411645646418743228

In GAP, Magma, Sage, TeX 

Dic_3\times C_2^2\times C_{10}
 
% in TeX
 
G:=Group("Dic3xC2^2xC10");
 
// GroupNames label
 
G:=SmallGroup(480,1163);
 
// by ID
 
G=gap.SmallGroup(480,1163);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,560,15686]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^10=d^6=1,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

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