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G = Dic5×C22×C6order 480 = 25·3·5

Direct product of C22×C6 and Dic5

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

Aliases: Dic5×C22×C6, C30.82C24, C53(C23×C12), C1512(C23×C4), (C23×C6).4D5, C24.3(C3×D5), C3012(C22×C4), C103(C22×C12), (C22×C30)⋊15C4, (C23×C10).5C6, (C23×C30).5C2, C6.82(C23×D5), C23.41(C6×D5), (C22×C10)⋊11C12, C10.14(C23×C6), (C2×C30).387C23, (C22×C6).138D10, (C22×C30).168C22, (C2×C30)⋊43(C2×C4), C2.2(D5×C22×C6), (C2×C10)⋊16(C2×C12), C22.33(D5×C2×C6), (C2×C10).69(C22×C6), (C22×C10).55(C2×C6), (C2×C6).380(C22×D5), SmallGroup(480,1148)

Series: Derived Chief Lower central Upper central

C1C5 — Dic5×C22×C6
C1C5C10C30C3×Dic5C6×Dic5C2×C6×Dic5 — Dic5×C22×C6
C5 — Dic5×C22×C6
C1C23×C6

Generators and relations for Dic5×C22×C6
 G = < a,b,c,d,e | a2=b2=c6=d10=1, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 880 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], C12 [×8], C2×C6 [×35], C15, C22×C4 [×14], C24, Dic5 [×8], C2×C10 [×35], C2×C12 [×28], C22×C6 [×15], C30, C30 [×14], C23×C4, C2×Dic5 [×28], C22×C10 [×15], C22×C12 [×14], C23×C6, C3×Dic5 [×8], C2×C30 [×35], C22×Dic5 [×14], C23×C10, C23×C12, C6×Dic5 [×28], C22×C30 [×15], C23×Dic5, C2×C6×Dic5 [×14], C23×C30, Dic5×C22×C6
Quotients: C1, C2 [×15], C3, C4 [×8], C22 [×35], C6 [×15], C2×C4 [×28], C23 [×15], D5, C12 [×8], C2×C6 [×35], C22×C4 [×14], C24, Dic5 [×8], D10 [×7], C2×C12 [×28], C22×C6 [×15], C3×D5, C23×C4, C2×Dic5 [×28], C22×D5 [×7], C22×C12 [×14], C23×C6, C3×Dic5 [×8], C6×D5 [×7], C22×Dic5 [×14], C23×D5, C23×C12, C6×Dic5 [×28], D5×C2×C6 [×7], C23×Dic5, C2×C6×Dic5 [×14], D5×C22×C6, Dic5×C22×C6

Smallest permutation representation of Dic5×C22×C6
Regular action on 480 points
Generators in S480
(1 214)(2 215)(3 216)(4 217)(5 218)(6 219)(7 220)(8 211)(9 212)(10 213)(11 226)(12 227)(13 228)(14 229)(15 230)(16 221)(17 222)(18 223)(19 224)(20 225)(21 236)(22 237)(23 238)(24 239)(25 240)(26 231)(27 232)(28 233)(29 234)(30 235)(31 186)(32 187)(33 188)(34 189)(35 190)(36 181)(37 182)(38 183)(39 184)(40 185)(41 196)(42 197)(43 198)(44 199)(45 200)(46 191)(47 192)(48 193)(49 194)(50 195)(51 206)(52 207)(53 208)(54 209)(55 210)(56 201)(57 202)(58 203)(59 204)(60 205)(61 156)(62 157)(63 158)(64 159)(65 160)(66 151)(67 152)(68 153)(69 154)(70 155)(71 166)(72 167)(73 168)(74 169)(75 170)(76 161)(77 162)(78 163)(79 164)(80 165)(81 176)(82 177)(83 178)(84 179)(85 180)(86 171)(87 172)(88 173)(89 174)(90 175)(91 126)(92 127)(93 128)(94 129)(95 130)(96 121)(97 122)(98 123)(99 124)(100 125)(101 136)(102 137)(103 138)(104 139)(105 140)(106 131)(107 132)(108 133)(109 134)(110 135)(111 146)(112 147)(113 148)(114 149)(115 150)(116 141)(117 142)(118 143)(119 144)(120 145)(241 456)(242 457)(243 458)(244 459)(245 460)(246 451)(247 452)(248 453)(249 454)(250 455)(251 466)(252 467)(253 468)(254 469)(255 470)(256 461)(257 462)(258 463)(259 464)(260 465)(261 476)(262 477)(263 478)(264 479)(265 480)(266 471)(267 472)(268 473)(269 474)(270 475)(271 426)(272 427)(273 428)(274 429)(275 430)(276 421)(277 422)(278 423)(279 424)(280 425)(281 436)(282 437)(283 438)(284 439)(285 440)(286 431)(287 432)(288 433)(289 434)(290 435)(291 446)(292 447)(293 448)(294 449)(295 450)(296 441)(297 442)(298 443)(299 444)(300 445)(301 396)(302 397)(303 398)(304 399)(305 400)(306 391)(307 392)(308 393)(309 394)(310 395)(311 406)(312 407)(313 408)(314 409)(315 410)(316 401)(317 402)(318 403)(319 404)(320 405)(321 416)(322 417)(323 418)(324 419)(325 420)(326 411)(327 412)(328 413)(329 414)(330 415)(331 366)(332 367)(333 368)(334 369)(335 370)(336 361)(337 362)(338 363)(339 364)(340 365)(341 376)(342 377)(343 378)(344 379)(345 380)(346 371)(347 372)(348 373)(349 374)(350 375)(351 386)(352 387)(353 388)(354 389)(355 390)(356 381)(357 382)(358 383)(359 384)(360 385)
(1 94)(2 95)(3 96)(4 97)(5 98)(6 99)(7 100)(8 91)(9 92)(10 93)(11 106)(12 107)(13 108)(14 109)(15 110)(16 101)(17 102)(18 103)(19 104)(20 105)(21 116)(22 117)(23 118)(24 119)(25 120)(26 111)(27 112)(28 113)(29 114)(30 115)(31 66)(32 67)(33 68)(34 69)(35 70)(36 61)(37 62)(38 63)(39 64)(40 65)(41 76)(42 77)(43 78)(44 79)(45 80)(46 71)(47 72)(48 73)(49 74)(50 75)(51 86)(52 87)(53 88)(54 89)(55 90)(56 81)(57 82)(58 83)(59 84)(60 85)(121 216)(122 217)(123 218)(124 219)(125 220)(126 211)(127 212)(128 213)(129 214)(130 215)(131 226)(132 227)(133 228)(134 229)(135 230)(136 221)(137 222)(138 223)(139 224)(140 225)(141 236)(142 237)(143 238)(144 239)(145 240)(146 231)(147 232)(148 233)(149 234)(150 235)(151 186)(152 187)(153 188)(154 189)(155 190)(156 181)(157 182)(158 183)(159 184)(160 185)(161 196)(162 197)(163 198)(164 199)(165 200)(166 191)(167 192)(168 193)(169 194)(170 195)(171 206)(172 207)(173 208)(174 209)(175 210)(176 201)(177 202)(178 203)(179 204)(180 205)(241 336)(242 337)(243 338)(244 339)(245 340)(246 331)(247 332)(248 333)(249 334)(250 335)(251 346)(252 347)(253 348)(254 349)(255 350)(256 341)(257 342)(258 343)(259 344)(260 345)(261 356)(262 357)(263 358)(264 359)(265 360)(266 351)(267 352)(268 353)(269 354)(270 355)(271 306)(272 307)(273 308)(274 309)(275 310)(276 301)(277 302)(278 303)(279 304)(280 305)(281 316)(282 317)(283 318)(284 319)(285 320)(286 311)(287 312)(288 313)(289 314)(290 315)(291 326)(292 327)(293 328)(294 329)(295 330)(296 321)(297 322)(298 323)(299 324)(300 325)(361 456)(362 457)(363 458)(364 459)(365 460)(366 451)(367 452)(368 453)(369 454)(370 455)(371 466)(372 467)(373 468)(374 469)(375 470)(376 461)(377 462)(378 463)(379 464)(380 465)(381 476)(382 477)(383 478)(384 479)(385 480)(386 471)(387 472)(388 473)(389 474)(390 475)(391 426)(392 427)(393 428)(394 429)(395 430)(396 421)(397 422)(398 423)(399 424)(400 425)(401 436)(402 437)(403 438)(404 439)(405 440)(406 431)(407 432)(408 433)(409 434)(410 435)(411 446)(412 447)(413 448)(414 449)(415 450)(416 441)(417 442)(418 443)(419 444)(420 445)
(1 49 29 39 19 59)(2 50 30 40 20 60)(3 41 21 31 11 51)(4 42 22 32 12 52)(5 43 23 33 13 53)(6 44 24 34 14 54)(7 45 25 35 15 55)(8 46 26 36 16 56)(9 47 27 37 17 57)(10 48 28 38 18 58)(61 101 81 91 71 111)(62 102 82 92 72 112)(63 103 83 93 73 113)(64 104 84 94 74 114)(65 105 85 95 75 115)(66 106 86 96 76 116)(67 107 87 97 77 117)(68 108 88 98 78 118)(69 109 89 99 79 119)(70 110 90 100 80 120)(121 161 141 151 131 171)(122 162 142 152 132 172)(123 163 143 153 133 173)(124 164 144 154 134 174)(125 165 145 155 135 175)(126 166 146 156 136 176)(127 167 147 157 137 177)(128 168 148 158 138 178)(129 169 149 159 139 179)(130 170 150 160 140 180)(181 221 201 211 191 231)(182 222 202 212 192 232)(183 223 203 213 193 233)(184 224 204 214 194 234)(185 225 205 215 195 235)(186 226 206 216 196 236)(187 227 207 217 197 237)(188 228 208 218 198 238)(189 229 209 219 199 239)(190 230 210 220 200 240)(241 281 261 271 251 291)(242 282 262 272 252 292)(243 283 263 273 253 293)(244 284 264 274 254 294)(245 285 265 275 255 295)(246 286 266 276 256 296)(247 287 267 277 257 297)(248 288 268 278 258 298)(249 289 269 279 259 299)(250 290 270 280 260 300)(301 341 321 331 311 351)(302 342 322 332 312 352)(303 343 323 333 313 353)(304 344 324 334 314 354)(305 345 325 335 315 355)(306 346 326 336 316 356)(307 347 327 337 317 357)(308 348 328 338 318 358)(309 349 329 339 319 359)(310 350 330 340 320 360)(361 401 381 391 371 411)(362 402 382 392 372 412)(363 403 383 393 373 413)(364 404 384 394 374 414)(365 405 385 395 375 415)(366 406 386 396 376 416)(367 407 387 397 377 417)(368 408 388 398 378 418)(369 409 389 399 379 419)(370 410 390 400 380 420)(421 461 441 451 431 471)(422 462 442 452 432 472)(423 463 443 453 433 473)(424 464 444 454 434 474)(425 465 445 455 435 475)(426 466 446 456 436 476)(427 467 447 457 437 477)(428 468 448 458 438 478)(429 469 449 459 439 479)(430 470 450 460 440 480)
(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)
(1 306 6 301)(2 305 7 310)(3 304 8 309)(4 303 9 308)(5 302 10 307)(11 314 16 319)(12 313 17 318)(13 312 18 317)(14 311 19 316)(15 320 20 315)(21 324 26 329)(22 323 27 328)(23 322 28 327)(24 321 29 326)(25 330 30 325)(31 334 36 339)(32 333 37 338)(33 332 38 337)(34 331 39 336)(35 340 40 335)(41 344 46 349)(42 343 47 348)(43 342 48 347)(44 341 49 346)(45 350 50 345)(51 354 56 359)(52 353 57 358)(53 352 58 357)(54 351 59 356)(55 360 60 355)(61 244 66 249)(62 243 67 248)(63 242 68 247)(64 241 69 246)(65 250 70 245)(71 254 76 259)(72 253 77 258)(73 252 78 257)(74 251 79 256)(75 260 80 255)(81 264 86 269)(82 263 87 268)(83 262 88 267)(84 261 89 266)(85 270 90 265)(91 274 96 279)(92 273 97 278)(93 272 98 277)(94 271 99 276)(95 280 100 275)(101 284 106 289)(102 283 107 288)(103 282 108 287)(104 281 109 286)(105 290 110 285)(111 294 116 299)(112 293 117 298)(113 292 118 297)(114 291 119 296)(115 300 120 295)(121 424 126 429)(122 423 127 428)(123 422 128 427)(124 421 129 426)(125 430 130 425)(131 434 136 439)(132 433 137 438)(133 432 138 437)(134 431 139 436)(135 440 140 435)(141 444 146 449)(142 443 147 448)(143 442 148 447)(144 441 149 446)(145 450 150 445)(151 454 156 459)(152 453 157 458)(153 452 158 457)(154 451 159 456)(155 460 160 455)(161 464 166 469)(162 463 167 468)(163 462 168 467)(164 461 169 466)(165 470 170 465)(171 474 176 479)(172 473 177 478)(173 472 178 477)(174 471 179 476)(175 480 180 475)(181 364 186 369)(182 363 187 368)(183 362 188 367)(184 361 189 366)(185 370 190 365)(191 374 196 379)(192 373 197 378)(193 372 198 377)(194 371 199 376)(195 380 200 375)(201 384 206 389)(202 383 207 388)(203 382 208 387)(204 381 209 386)(205 390 210 385)(211 394 216 399)(212 393 217 398)(213 392 218 397)(214 391 219 396)(215 400 220 395)(221 404 226 409)(222 403 227 408)(223 402 228 407)(224 401 229 406)(225 410 230 405)(231 414 236 419)(232 413 237 418)(233 412 238 417)(234 411 239 416)(235 420 240 415)

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

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

192 conjugacy classes

class 1 2A···2O3A3B4A···4P5A5B6A···6AD10A···10AD12A···12AF15A15B15C15D30A···30BH
order12···2334···4556···610···1012···121515151530···30
size11···1115···5221···12···25···522222···2

192 irreducible representations

dim11111111222222
type++++-+
imageC1C2C2C3C4C6C6C12D5Dic5D10C3×D5C3×Dic5C6×D5
kernelDic5×C22×C6C2×C6×Dic5C23×C30C23×Dic5C22×C30C22×Dic5C23×C10C22×C10C23×C6C22×C6C22×C6C24C23C23
# reps1141216282322161443228

Matrix representation of Dic5×C22×C6 in GL5(𝔽61)

600000
01000
00100
000600
000060
,
600000
060000
00100
00010
00001
,
600000
01000
006000
000140
000014
,
10000
060000
00100
00001
0006017
,
10000
050000
00100
0002034
000841

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60],[60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,14,0,0,0,0,0,14],[1,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,1,17],[1,0,0,0,0,0,50,0,0,0,0,0,1,0,0,0,0,0,20,8,0,0,0,34,41] >;

Dic5×C22×C6 in GAP, Magma, Sage, TeX

{\rm Dic}_5\times C_2^2\times C_6
% in TeX

G:=Group("Dic5xC2^2xC6");
// GroupNames label

G:=SmallGroup(480,1148);
// by ID

G=gap.SmallGroup(480,1148);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-5,336,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^6=d^10=1,e^2=d^5,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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