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G = C3×Dic5⋊Q8order 480 = 25·3·5

Direct product of C3 and Dic5⋊Q8

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

Aliases: C3×Dic5⋊Q8, C60.131D4, C1517(C4⋊Q8), (C6×Q8).9D5, C6.53(Q8×D5), Dic52(C3×Q8), (C3×Dic5)⋊9Q8, C20.21(C3×D4), C10.56(C6×D4), (Q8×C10).8C6, (Q8×C30).9C2, C10.15(C6×Q8), C30.413(C2×D4), C30.105(C2×Q8), (C4×Dic5).3C6, (C2×C12).243D10, C12.78(C5⋊D4), C10.D4.6C6, (C2×C60).425C22, (C2×C30).373C23, (C12×Dic5).13C2, (C6×Dic10).21C2, (C2×Dic10).10C6, (C6×Dic5).253C22, C53(C3×C4⋊Q8), C2.8(C3×Q8×D5), (C2×C4).54(C6×D5), C4.10(C3×C5⋊D4), C2.20(C6×C5⋊D4), (C2×Q8).6(C3×D5), C22.63(D5×C2×C6), (C2×C20).63(C2×C6), C6.141(C2×C5⋊D4), (C2×C10).56(C22×C6), (C2×Dic5).17(C2×C6), (C2×C6).369(C22×D5), (C3×C10.D4).16C2, SmallGroup(480,737)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C3×Dic5⋊Q8
C1C5C10C2×C10C2×C30C6×Dic5C12×Dic5 — C3×Dic5⋊Q8
C5C2×C10 — C3×Dic5⋊Q8
C1C2×C6C6×Q8

Generators and relations for C3×Dic5⋊Q8
 G = < a,b,c,d,e | a3=b10=d4=1, c2=b5, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b5c, ce=ec, ede-1=d-1 >

Subgroups: 352 in 136 conjugacy classes, 74 normal (26 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×8], C22, C5, C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×4], Q8 [×4], C10, C10 [×2], C12 [×2], C12 [×8], C2×C6, C15, C42, C4⋊C4 [×4], C2×Q8, C2×Q8, Dic5 [×4], Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C2×C12, C2×C12 [×2], C2×C12 [×4], C3×Q8 [×4], C30, C30 [×2], C4⋊Q8, Dic10 [×2], C2×Dic5 [×4], C2×C20, C2×C20 [×2], C5×Q8 [×2], C4×C12, C3×C4⋊C4 [×4], C6×Q8, C6×Q8, C3×Dic5 [×4], C3×Dic5 [×2], C60 [×2], C60 [×2], C2×C30, C4×Dic5, C10.D4 [×4], C2×Dic10, Q8×C10, C3×C4⋊Q8, C3×Dic10 [×2], C6×Dic5 [×4], C2×C60, C2×C60 [×2], Q8×C15 [×2], Dic5⋊Q8, C12×Dic5, C3×C10.D4 [×4], C6×Dic10, Q8×C30, C3×Dic5⋊Q8
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], Q8 [×4], C23, D5, C2×C6 [×7], C2×D4, C2×Q8 [×2], D10 [×3], C3×D4 [×2], C3×Q8 [×4], C22×C6, C3×D5, C4⋊Q8, C5⋊D4 [×2], C22×D5, C6×D4, C6×Q8 [×2], C6×D5 [×3], Q8×D5 [×2], C2×C5⋊D4, C3×C4⋊Q8, C3×C5⋊D4 [×2], D5×C2×C6, Dic5⋊Q8, C3×Q8×D5 [×2], C6×C5⋊D4, C3×Dic5⋊Q8

Smallest permutation representation of C3×Dic5⋊Q8
Regular action on 480 points
Generators in S480
(1 62 42)(2 63 43)(3 64 44)(4 65 45)(5 66 46)(6 67 47)(7 68 48)(8 69 49)(9 70 50)(10 61 41)(11 468 448)(12 469 449)(13 470 450)(14 461 441)(15 462 442)(16 463 443)(17 464 444)(18 465 445)(19 466 446)(20 467 447)(21 59 39)(22 60 40)(23 51 31)(24 52 32)(25 53 33)(26 54 34)(27 55 35)(28 56 36)(29 57 37)(30 58 38)(71 111 91)(72 112 92)(73 113 93)(74 114 94)(75 115 95)(76 116 96)(77 117 97)(78 118 98)(79 119 99)(80 120 100)(81 121 101)(82 122 102)(83 123 103)(84 124 104)(85 125 105)(86 126 106)(87 127 107)(88 128 108)(89 129 109)(90 130 110)(131 171 151)(132 172 152)(133 173 153)(134 174 154)(135 175 155)(136 176 156)(137 177 157)(138 178 158)(139 179 159)(140 180 160)(141 181 161)(142 182 162)(143 183 163)(144 184 164)(145 185 165)(146 186 166)(147 187 167)(148 188 168)(149 189 169)(150 190 170)(191 231 211)(192 232 212)(193 233 213)(194 234 214)(195 235 215)(196 236 216)(197 237 217)(198 238 218)(199 239 219)(200 240 220)(201 244 221)(202 245 222)(203 246 223)(204 247 224)(205 248 225)(206 249 226)(207 250 227)(208 241 228)(209 242 229)(210 243 230)(251 291 271)(252 292 272)(253 293 273)(254 294 274)(255 295 275)(256 296 276)(257 297 277)(258 298 278)(259 299 279)(260 300 280)(261 301 281)(262 302 282)(263 303 283)(264 304 284)(265 305 285)(266 306 286)(267 307 287)(268 308 288)(269 309 289)(270 310 290)(311 351 331)(312 352 332)(313 353 333)(314 354 334)(315 355 335)(316 356 336)(317 357 337)(318 358 338)(319 359 339)(320 360 340)(321 361 341)(322 362 342)(323 363 343)(324 364 344)(325 365 345)(326 366 346)(327 367 347)(328 368 348)(329 369 349)(330 370 350)(371 411 391)(372 412 392)(373 413 393)(374 414 394)(375 415 395)(376 416 396)(377 417 397)(378 418 398)(379 419 399)(380 420 400)(381 421 401)(382 422 402)(383 423 403)(384 424 404)(385 425 405)(386 426 406)(387 427 407)(388 428 408)(389 429 409)(390 430 410)(431 471 451)(432 472 452)(433 473 453)(434 474 454)(435 475 455)(436 476 456)(437 477 457)(438 478 458)(439 479 459)(440 480 460)
(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)
(1 379 6 374)(2 378 7 373)(3 377 8 372)(4 376 9 371)(5 375 10 380)(11 111 16 116)(12 120 17 115)(13 119 18 114)(14 118 19 113)(15 117 20 112)(21 385 26 390)(22 384 27 389)(23 383 28 388)(24 382 29 387)(25 381 30 386)(31 403 36 408)(32 402 37 407)(33 401 38 406)(34 410 39 405)(35 409 40 404)(41 400 46 395)(42 399 47 394)(43 398 48 393)(44 397 49 392)(45 396 50 391)(51 423 56 428)(52 422 57 427)(53 421 58 426)(54 430 59 425)(55 429 60 424)(61 420 66 415)(62 419 67 414)(63 418 68 413)(64 417 69 412)(65 416 70 411)(71 443 76 448)(72 442 77 447)(73 441 78 446)(74 450 79 445)(75 449 80 444)(81 440 86 435)(82 439 87 434)(83 438 88 433)(84 437 89 432)(85 436 90 431)(91 463 96 468)(92 462 97 467)(93 461 98 466)(94 470 99 465)(95 469 100 464)(101 460 106 455)(102 459 107 454)(103 458 108 453)(104 457 109 452)(105 456 110 451)(121 480 126 475)(122 479 127 474)(123 478 128 473)(124 477 129 472)(125 476 130 471)(131 259 136 254)(132 258 137 253)(133 257 138 252)(134 256 139 251)(135 255 140 260)(141 264 146 269)(142 263 147 268)(143 262 148 267)(144 261 149 266)(145 270 150 265)(151 279 156 274)(152 278 157 273)(153 277 158 272)(154 276 159 271)(155 275 160 280)(161 284 166 289)(162 283 167 288)(163 282 168 287)(164 281 169 286)(165 290 170 285)(171 299 176 294)(172 298 177 293)(173 297 178 292)(174 296 179 291)(175 295 180 300)(181 304 186 309)(182 303 187 308)(183 302 188 307)(184 301 189 306)(185 310 190 305)(191 319 196 314)(192 318 197 313)(193 317 198 312)(194 316 199 311)(195 315 200 320)(201 324 206 329)(202 323 207 328)(203 322 208 327)(204 321 209 326)(205 330 210 325)(211 339 216 334)(212 338 217 333)(213 337 218 332)(214 336 219 331)(215 335 220 340)(221 344 226 349)(222 343 227 348)(223 342 228 347)(224 341 229 346)(225 350 230 345)(231 359 236 354)(232 358 237 353)(233 357 238 352)(234 356 239 351)(235 355 240 360)(241 367 246 362)(242 366 247 361)(243 365 248 370)(244 364 249 369)(245 363 250 368)
(1 71 23 82)(2 72 24 83)(3 73 25 84)(4 74 26 85)(5 75 27 86)(6 76 28 87)(7 77 29 88)(8 78 30 89)(9 79 21 90)(10 80 22 81)(11 423 474 419)(12 424 475 420)(13 425 476 411)(14 426 477 412)(15 427 478 413)(16 428 479 414)(17 429 480 415)(18 430 471 416)(19 421 472 417)(20 422 473 418)(31 102 42 91)(32 103 43 92)(33 104 44 93)(34 105 45 94)(35 106 46 95)(36 107 47 96)(37 108 48 97)(38 109 49 98)(39 110 50 99)(40 101 41 100)(51 122 62 111)(52 123 63 112)(53 124 64 113)(54 125 65 114)(55 126 66 115)(56 127 67 116)(57 128 68 117)(58 129 69 118)(59 130 70 119)(60 121 61 120)(131 191 142 202)(132 192 143 203)(133 193 144 204)(134 194 145 205)(135 195 146 206)(136 196 147 207)(137 197 148 208)(138 198 149 209)(139 199 150 210)(140 200 141 201)(151 211 162 222)(152 212 163 223)(153 213 164 224)(154 214 165 225)(155 215 166 226)(156 216 167 227)(157 217 168 228)(158 218 169 229)(159 219 170 230)(160 220 161 221)(171 231 182 245)(172 232 183 246)(173 233 184 247)(174 234 185 248)(175 235 186 249)(176 236 187 250)(177 237 188 241)(178 238 189 242)(179 239 190 243)(180 240 181 244)(251 316 265 330)(252 317 266 321)(253 318 267 322)(254 319 268 323)(255 320 269 324)(256 311 270 325)(257 312 261 326)(258 313 262 327)(259 314 263 328)(260 315 264 329)(271 336 285 350)(272 337 286 341)(273 338 287 342)(274 339 288 343)(275 340 289 344)(276 331 290 345)(277 332 281 346)(278 333 282 347)(279 334 283 348)(280 335 284 349)(291 356 305 370)(292 357 306 361)(293 358 307 362)(294 359 308 363)(295 360 309 364)(296 351 310 365)(297 352 301 366)(298 353 302 367)(299 354 303 368)(300 355 304 369)(371 450 385 436)(372 441 386 437)(373 442 387 438)(374 443 388 439)(375 444 389 440)(376 445 390 431)(377 446 381 432)(378 447 382 433)(379 448 383 434)(380 449 384 435)(391 470 405 456)(392 461 406 457)(393 462 407 458)(394 463 408 459)(395 464 409 460)(396 465 410 451)(397 466 401 452)(398 467 402 453)(399 468 403 454)(400 469 404 455)
(1 131 23 142)(2 132 24 143)(3 133 25 144)(4 134 26 145)(5 135 27 146)(6 136 28 147)(7 137 29 148)(8 138 30 149)(9 139 21 150)(10 140 22 141)(11 368 474 354)(12 369 475 355)(13 370 476 356)(14 361 477 357)(15 362 478 358)(16 363 479 359)(17 364 480 360)(18 365 471 351)(19 366 472 352)(20 367 473 353)(31 162 42 151)(32 163 43 152)(33 164 44 153)(34 165 45 154)(35 166 46 155)(36 167 47 156)(37 168 48 157)(38 169 49 158)(39 170 50 159)(40 161 41 160)(51 182 62 171)(52 183 63 172)(53 184 64 173)(54 185 65 174)(55 186 66 175)(56 187 67 176)(57 188 68 177)(58 189 69 178)(59 190 70 179)(60 181 61 180)(71 202 82 191)(72 203 83 192)(73 204 84 193)(74 205 85 194)(75 206 86 195)(76 207 87 196)(77 208 88 197)(78 209 89 198)(79 210 90 199)(80 201 81 200)(91 222 102 211)(92 223 103 212)(93 224 104 213)(94 225 105 214)(95 226 106 215)(96 227 107 216)(97 228 108 217)(98 229 109 218)(99 230 110 219)(100 221 101 220)(111 245 122 231)(112 246 123 232)(113 247 124 233)(114 248 125 234)(115 249 126 235)(116 250 127 236)(117 241 128 237)(118 242 129 238)(119 243 130 239)(120 244 121 240)(251 385 265 371)(252 386 266 372)(253 387 267 373)(254 388 268 374)(255 389 269 375)(256 390 270 376)(257 381 261 377)(258 382 262 378)(259 383 263 379)(260 384 264 380)(271 405 285 391)(272 406 286 392)(273 407 287 393)(274 408 288 394)(275 409 289 395)(276 410 290 396)(277 401 281 397)(278 402 282 398)(279 403 283 399)(280 404 284 400)(291 425 305 411)(292 426 306 412)(293 427 307 413)(294 428 308 414)(295 429 309 415)(296 430 310 416)(297 421 301 417)(298 422 302 418)(299 423 303 419)(300 424 304 420)(311 445 325 431)(312 446 326 432)(313 447 327 433)(314 448 328 434)(315 449 329 435)(316 450 330 436)(317 441 321 437)(318 442 322 438)(319 443 323 439)(320 444 324 440)(331 465 345 451)(332 466 346 452)(333 467 347 453)(334 468 348 454)(335 469 349 455)(336 470 350 456)(337 461 341 457)(338 462 342 458)(339 463 343 459)(340 464 344 460)

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

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

102 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F4G4H4I4J5A5B6A···6F10A···10F12A12B12C12D12E12F12G12H12I···12P12Q12R12S12T15A15B15C15D20A···20L30A···30L60A···60X
order1222334444444444556···610···10121212121212121212···12121212121515151520···2030···3060···60
size1111112244101010102020221···12···22222444410···102020202022224···42···24···4

102 irreducible representations

dim1111111111222222222244
type+++++-+++-
imageC1C2C2C2C2C3C6C6C6C6Q8D4D5D10C3×Q8C3×D4C3×D5C5⋊D4C6×D5C3×C5⋊D4Q8×D5C3×Q8×D5
kernelC3×Dic5⋊Q8C12×Dic5C3×C10.D4C6×Dic10Q8×C30Dic5⋊Q8C4×Dic5C10.D4C2×Dic10Q8×C10C3×Dic5C60C6×Q8C2×C12Dic5C20C2×Q8C12C2×C4C4C6C2
# reps114112282242268448121648

Matrix representation of C3×Dic5⋊Q8 in GL4(𝔽61) generated by

47000
04700
00470
00047
,
1000
0100
00181
00600
,
60000
06000
00359
00558
,
602000
6100
003117
004430
,
174000
84400
0010
0001
G:=sub<GL(4,GF(61))| [47,0,0,0,0,47,0,0,0,0,47,0,0,0,0,47],[1,0,0,0,0,1,0,0,0,0,18,60,0,0,1,0],[60,0,0,0,0,60,0,0,0,0,3,5,0,0,59,58],[60,6,0,0,20,1,0,0,0,0,31,44,0,0,17,30],[17,8,0,0,40,44,0,0,0,0,1,0,0,0,0,1] >;

C3×Dic5⋊Q8 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_5\rtimes Q_8
% in TeX

G:=Group("C3xDic5:Q8");
// GroupNames label

G:=SmallGroup(480,737);
// by ID

G=gap.SmallGroup(480,737);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,176,1094,303,142,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^10=d^4=1,c^2=b^5,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^5*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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