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G = Dic3×C22×C10order 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, C32(C23×C20), (C22×C6)⋊5C20, C62(C22×C20), C1513(C23×C4), C24.3(C5×S3), (C22×C30)⋊17C4, C3013(C22×C4), (C23×C6).3C10, (C23×C10).6S3, (C23×C30).7C2, C6.14(C23×C10), C10.82(S3×C23), C23.41(S3×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), (C2×C6).69(C22×C10), (C22×C6).47(C2×C10), (C2×C10).380(C22×S3), SmallGroup(480,1163)

Series: Derived Chief Lower central Upper central

C1C3 — Dic3×C22×C10
C1C3C6C30C5×Dic3C10×Dic3Dic3×C2×C10 — Dic3×C22×C10
C3 — Dic3×C22×C10
C1C23×C10

Generators and relations for Dic3×C22×C10
 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 >

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

Smallest permutation representation of Dic3×C22×C10
Regular action on 480 points
Generators in S480
(1 290)(2 281)(3 282)(4 283)(5 284)(6 285)(7 286)(8 287)(9 288)(10 289)(11 325)(12 326)(13 327)(14 328)(15 329)(16 330)(17 321)(18 322)(19 323)(20 324)(21 300)(22 291)(23 292)(24 293)(25 294)(26 295)(27 296)(28 297)(29 298)(30 299)(31 333)(32 334)(33 335)(34 336)(35 337)(36 338)(37 339)(38 340)(39 331)(40 332)(41 301)(42 302)(43 303)(44 304)(45 305)(46 306)(47 307)(48 308)(49 309)(50 310)(51 311)(52 312)(53 313)(54 314)(55 315)(56 316)(57 317)(58 318)(59 319)(60 320)(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)(81 274)(82 275)(83 276)(84 277)(85 278)(86 279)(87 280)(88 271)(89 272)(90 273)(91 268)(92 269)(93 270)(94 261)(95 262)(96 263)(97 264)(98 265)(99 266)(100 267)(101 237)(102 238)(103 239)(104 240)(105 231)(106 232)(107 233)(108 234)(109 235)(110 236)(111 225)(112 226)(113 227)(114 228)(115 229)(116 230)(117 221)(118 222)(119 223)(120 224)(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)(141 214)(142 215)(143 216)(144 217)(145 218)(146 219)(147 220)(148 211)(149 212)(150 213)(151 208)(152 209)(153 210)(154 201)(155 202)(156 203)(157 204)(158 205)(159 206)(160 207)(161 177)(162 178)(163 179)(164 180)(165 171)(166 172)(167 173)(168 174)(169 175)(170 176)(341 477)(342 478)(343 479)(344 480)(345 471)(346 472)(347 473)(348 474)(349 475)(350 476)(351 465)(352 466)(353 467)(354 468)(355 469)(356 470)(357 461)(358 462)(359 463)(360 464)(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)(381 454)(382 455)(383 456)(384 457)(385 458)(386 459)(387 460)(388 451)(389 452)(390 453)(391 448)(392 449)(393 450)(394 441)(395 442)(396 443)(397 444)(398 445)(399 446)(400 447)(401 417)(402 418)(403 419)(404 420)(405 411)(406 412)(407 413)(408 414)(409 415)(410 416)
(1 170)(2 161)(3 162)(4 163)(5 164)(6 165)(7 166)(8 167)(9 168)(10 169)(11 445)(12 446)(13 447)(14 448)(15 449)(16 450)(17 441)(18 442)(19 443)(20 444)(21 420)(22 411)(23 412)(24 413)(25 414)(26 415)(27 416)(28 417)(29 418)(30 419)(31 453)(32 454)(33 455)(34 456)(35 457)(36 458)(37 459)(38 460)(39 451)(40 452)(41 421)(42 422)(43 423)(44 424)(45 425)(46 426)(47 427)(48 428)(49 429)(50 430)(51 431)(52 432)(53 433)(54 434)(55 435)(56 436)(57 437)(58 438)(59 439)(60 440)(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)(81 154)(82 155)(83 156)(84 157)(85 158)(86 159)(87 160)(88 151)(89 152)(90 153)(91 148)(92 149)(93 150)(94 141)(95 142)(96 143)(97 144)(98 145)(99 146)(100 147)(101 117)(102 118)(103 119)(104 120)(105 111)(106 112)(107 113)(108 114)(109 115)(110 116)(171 285)(172 286)(173 287)(174 288)(175 289)(176 290)(177 281)(178 282)(179 283)(180 284)(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)(201 274)(202 275)(203 276)(204 277)(205 278)(206 279)(207 280)(208 271)(209 272)(210 273)(211 268)(212 269)(213 270)(214 261)(215 262)(216 263)(217 264)(218 265)(219 266)(220 267)(221 237)(222 238)(223 239)(224 240)(225 231)(226 232)(227 233)(228 234)(229 235)(230 236)(291 405)(292 406)(293 407)(294 408)(295 409)(296 410)(297 401)(298 402)(299 403)(300 404)(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)(321 394)(322 395)(323 396)(324 397)(325 398)(326 399)(327 400)(328 391)(329 392)(330 393)(331 388)(332 389)(333 390)(334 381)(335 382)(336 383)(337 384)(338 385)(339 386)(340 387)(341 357)(342 358)(343 359)(344 360)(345 351)(346 352)(347 353)(348 354)(349 355)(350 356)(461 477)(462 478)(463 479)(464 480)(465 471)(466 472)(467 473)(468 474)(469 475)(470 476)
(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 97 105 61 89)(2 76 98 106 62 90)(3 77 99 107 63 81)(4 78 100 108 64 82)(5 79 91 109 65 83)(6 80 92 110 66 84)(7 71 93 101 67 85)(8 72 94 102 68 86)(9 73 95 103 69 87)(10 74 96 104 70 88)(11 47 477 31 51 23)(12 48 478 32 52 24)(13 49 479 33 53 25)(14 50 480 34 54 26)(15 41 471 35 55 27)(16 42 472 36 56 28)(17 43 473 37 57 29)(18 44 474 38 58 30)(19 45 475 39 59 21)(20 46 476 40 60 22)(111 140 152 170 126 144)(112 131 153 161 127 145)(113 132 154 162 128 146)(114 133 155 163 129 147)(115 134 156 164 130 148)(116 135 157 165 121 149)(117 136 158 166 122 150)(118 137 159 167 123 141)(119 138 160 168 124 142)(120 139 151 169 125 143)(171 200 212 230 186 204)(172 191 213 221 187 205)(173 192 214 222 188 206)(174 193 215 223 189 207)(175 194 216 224 190 208)(176 195 217 225 181 209)(177 196 218 226 182 210)(178 197 219 227 183 201)(179 198 220 228 184 202)(180 199 211 229 185 203)(231 260 272 290 246 264)(232 251 273 281 247 265)(233 252 274 282 248 266)(234 253 275 283 249 267)(235 254 276 284 250 268)(236 255 277 285 241 269)(237 256 278 286 242 270)(238 257 279 287 243 261)(239 258 280 288 244 262)(240 259 271 289 245 263)(291 324 306 350 332 320)(292 325 307 341 333 311)(293 326 308 342 334 312)(294 327 309 343 335 313)(295 328 310 344 336 314)(296 329 301 345 337 315)(297 330 302 346 338 316)(298 321 303 347 339 317)(299 322 304 348 340 318)(300 323 305 349 331 319)(351 384 366 410 392 380)(352 385 367 401 393 371)(353 386 368 402 394 372)(354 387 369 403 395 373)(355 388 370 404 396 374)(356 389 361 405 397 375)(357 390 362 406 398 376)(358 381 363 407 399 377)(359 382 364 408 400 378)(360 383 365 409 391 379)(411 444 426 470 452 440)(412 445 427 461 453 431)(413 446 428 462 454 432)(414 447 429 463 455 433)(415 448 430 464 456 434)(416 449 421 465 457 435)(417 450 422 466 458 436)(418 441 423 467 459 437)(419 442 424 468 460 438)(420 443 425 469 451 439)
(1 405 105 356)(2 406 106 357)(3 407 107 358)(4 408 108 359)(5 409 109 360)(6 410 110 351)(7 401 101 352)(8 402 102 353)(9 403 103 354)(10 404 104 355)(11 218 31 210)(12 219 32 201)(13 220 33 202)(14 211 34 203)(15 212 35 204)(16 213 36 205)(17 214 37 206)(18 215 38 207)(19 216 39 208)(20 217 40 209)(21 224 475 175)(22 225 476 176)(23 226 477 177)(24 227 478 178)(25 228 479 179)(26 229 480 180)(27 230 471 171)(28 221 472 172)(29 222 473 173)(30 223 474 174)(41 200 55 186)(42 191 56 187)(43 192 57 188)(44 193 58 189)(45 194 59 190)(46 195 60 181)(47 196 51 182)(48 197 52 183)(49 198 53 184)(50 199 54 185)(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)(81 399 99 381)(82 400 100 382)(83 391 91 383)(84 392 92 384)(85 393 93 385)(86 394 94 386)(87 395 95 387)(88 396 96 388)(89 397 97 389)(90 398 98 390)(111 350 170 291)(112 341 161 292)(113 342 162 293)(114 343 163 294)(115 344 164 295)(116 345 165 296)(117 346 166 297)(118 347 167 298)(119 348 168 299)(120 349 169 300)(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)(141 339 159 321)(142 340 160 322)(143 331 151 323)(144 332 152 324)(145 333 153 325)(146 334 154 326)(147 335 155 327)(148 336 156 328)(149 337 157 329)(150 338 158 330)(231 470 290 411)(232 461 281 412)(233 462 282 413)(234 463 283 414)(235 464 284 415)(236 465 285 416)(237 466 286 417)(238 467 287 418)(239 468 288 419)(240 469 289 420)(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)(261 459 279 441)(262 460 280 442)(263 451 271 443)(264 452 272 444)(265 453 273 445)(266 454 274 446)(267 455 275 447)(268 456 276 448)(269 457 277 449)(270 458 278 450)

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

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

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

Matrix representation of Dic3×C22×C10 in 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] >;

Dic3×C22×C10 in GAP, Magma, Sage, TeX

{\rm 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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