direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×Dic3.Q8, Dic3.(C5×Q8), C10.49(S3×Q8), C6.11(Q8×C10), (C2×C20).274D6, C4⋊Dic3.6C10, C30.109(C2×Q8), (C5×Dic3).4Q8, Dic3⋊C4.3C10, (C4×Dic3).9C10, C15⋊20(C42.C2), C30.205(C4○D4), (C2×C60).332C22, (C2×C30).409C23, (Dic3×C20).20C2, C10.118(C4○D12), C10.114(D4⋊2S3), (C10×Dic3).143C22, C2.5(C5×S3×Q8), C4⋊C4.5(C5×S3), (C3×C4⋊C4).6C10, (C5×C4⋊C4).12S3, C3⋊2(C5×C42.C2), (C2×C4).9(S3×C10), C6.10(C5×C4○D4), (C15×C4⋊C4).20C2, C2.13(C5×C4○D12), C22.47(S3×C2×C10), (C2×C12).61(C2×C10), C2.11(C5×D4⋊2S3), (C5×C4⋊Dic3).20C2, (C5×Dic3⋊C4).15C2, (C2×C6).30(C22×C10), (C2×Dic3).9(C2×C10), (C2×C10).343(C22×S3), SmallGroup(480,768)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Dic3.Q8
G = < a,b,c,d,e | a5=b6=d4=1, c2=b3, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=b3c, ce=ec, ede-1=b3d-1 >
Subgroups: 212 in 112 conjugacy classes, 62 normal (58 characteristic)
C1, C2, C3, C4, C22, C5, C6, C2×C4, C2×C4, C10, Dic3, Dic3, C12, C2×C6, C15, C42, C4⋊C4, C4⋊C4, C20, C2×C10, C2×Dic3, C2×C12, C30, C42.C2, C2×C20, C2×C20, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C5×Dic3, C5×Dic3, C60, C2×C30, C4×C20, C5×C4⋊C4, C5×C4⋊C4, Dic3.Q8, C10×Dic3, C2×C60, C5×C42.C2, Dic3×C20, C5×Dic3⋊C4, C5×C4⋊Dic3, C15×C4⋊C4, C5×Dic3.Q8
Quotients: C1, C2, C22, C5, S3, Q8, C23, C10, D6, C2×Q8, C4○D4, C2×C10, C22×S3, C5×S3, C42.C2, C5×Q8, C22×C10, C4○D12, D4⋊2S3, S3×Q8, S3×C10, Q8×C10, C5×C4○D4, Dic3.Q8, S3×C2×C10, C5×C42.C2, C5×C4○D12, C5×D4⋊2S3, C5×S3×Q8, C5×Dic3.Q8
►Regular action on 480 points
Generators in S480
(1 62 50 38 26)(2 63 51 39 27)(3 64 52 40 28)(4 65 53 41 29)(5 66 54 42 30)(6 61 49 37 25)(7 472 460 448 436)(8 473 461 449 437)(9 474 462 450 438)(10 469 457 445 433)(11 470 458 446 434)(12 471 459 447 435)(13 59 47 35 23)(14 60 48 36 24)(15 55 43 31 19)(16 56 44 32 20)(17 57 45 33 21)(18 58 46 34 22)(67 115 103 91 79)(68 116 104 92 80)(69 117 105 93 81)(70 118 106 94 82)(71 119 107 95 83)(72 120 108 96 84)(73 121 109 97 85)(74 122 110 98 86)(75 123 111 99 87)(76 124 112 100 88)(77 125 113 101 89)(78 126 114 102 90)(127 175 163 151 139)(128 176 164 152 140)(129 177 165 153 141)(130 178 166 154 142)(131 179 167 155 143)(132 180 168 156 144)(133 181 169 157 145)(134 182 170 158 146)(135 183 171 159 147)(136 184 172 160 148)(137 185 173 161 149)(138 186 174 162 150)(187 235 223 211 199)(188 236 224 212 200)(189 237 225 213 201)(190 238 226 214 202)(191 239 227 215 203)(192 240 228 216 204)(193 242 229 217 205)(194 243 230 218 206)(195 244 231 219 207)(196 245 232 220 208)(197 246 233 221 209)(198 241 234 222 210)(247 295 283 271 259)(248 296 284 272 260)(249 297 285 273 261)(250 298 286 274 262)(251 299 287 275 263)(252 300 288 276 264)(253 301 289 277 265)(254 302 290 278 266)(255 303 291 279 267)(256 304 292 280 268)(257 305 293 281 269)(258 306 294 282 270)(307 355 343 331 319)(308 356 344 332 320)(309 357 345 333 321)(310 358 346 334 322)(311 359 347 335 323)(312 360 348 336 324)(313 361 349 337 325)(314 362 350 338 326)(315 363 351 339 327)(316 364 352 340 328)(317 365 353 341 329)(318 366 354 342 330)(367 415 403 391 379)(368 416 404 392 380)(369 417 405 393 381)(370 418 406 394 382)(371 419 407 395 383)(372 420 408 396 384)(373 421 409 397 385)(374 422 410 398 386)(375 423 411 399 387)(376 424 412 400 388)(377 425 413 401 389)(378 426 414 402 390)(427 475 463 451 439)(428 476 464 452 440)(429 477 465 453 441)(430 478 466 454 442)(431 479 467 455 443)(432 480 468 456 444)
(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)
(1 371 4 368)(2 370 5 367)(3 369 6 372)(7 115 10 118)(8 120 11 117)(9 119 12 116)(13 375 16 378)(14 374 17 377)(15 373 18 376)(19 385 22 388)(20 390 23 387)(21 389 24 386)(25 384 28 381)(26 383 29 380)(27 382 30 379)(31 397 34 400)(32 402 35 399)(33 401 36 398)(37 396 40 393)(38 395 41 392)(39 394 42 391)(43 409 46 412)(44 414 47 411)(45 413 48 410)(49 408 52 405)(50 407 53 404)(51 406 54 403)(55 421 58 424)(56 426 59 423)(57 425 60 422)(61 420 64 417)(62 419 65 416)(63 418 66 415)(67 433 70 436)(68 438 71 435)(69 437 72 434)(73 432 76 429)(74 431 77 428)(75 430 78 427)(79 445 82 448)(80 450 83 447)(81 449 84 446)(85 444 88 441)(86 443 89 440)(87 442 90 439)(91 457 94 460)(92 462 95 459)(93 461 96 458)(97 456 100 453)(98 455 101 452)(99 454 102 451)(103 469 106 472)(104 474 107 471)(105 473 108 470)(109 468 112 465)(110 467 113 464)(111 466 114 463)(121 480 124 477)(122 479 125 476)(123 478 126 475)(127 251 130 248)(128 250 131 247)(129 249 132 252)(133 254 136 257)(134 253 137 256)(135 258 138 255)(139 263 142 260)(140 262 143 259)(141 261 144 264)(145 266 148 269)(146 265 149 268)(147 270 150 267)(151 275 154 272)(152 274 155 271)(153 273 156 276)(157 278 160 281)(158 277 161 280)(159 282 162 279)(163 287 166 284)(164 286 167 283)(165 285 168 288)(169 290 172 293)(170 289 173 292)(171 294 174 291)(175 299 178 296)(176 298 179 295)(177 297 180 300)(181 302 184 305)(182 301 185 304)(183 306 186 303)(187 311 190 308)(188 310 191 307)(189 309 192 312)(193 314 196 317)(194 313 197 316)(195 318 198 315)(199 323 202 320)(200 322 203 319)(201 321 204 324)(205 326 208 329)(206 325 209 328)(207 330 210 327)(211 335 214 332)(212 334 215 331)(213 333 216 336)(217 338 220 341)(218 337 221 340)(219 342 222 339)(223 347 226 344)(224 346 227 343)(225 345 228 348)(229 350 232 353)(230 349 233 352)(231 354 234 351)(235 359 238 356)(236 358 239 355)(237 357 240 360)(241 363 244 366)(242 362 245 365)(243 361 246 364)
(1 70 18 74)(2 71 13 75)(3 72 14 76)(4 67 15 77)(5 68 16 78)(6 69 17 73)(7 421 479 416)(8 422 480 417)(9 423 475 418)(10 424 476 419)(11 425 477 420)(12 426 478 415)(19 89 29 79)(20 90 30 80)(21 85 25 81)(22 86 26 82)(23 87 27 83)(24 88 28 84)(31 101 41 91)(32 102 42 92)(33 97 37 93)(34 98 38 94)(35 99 39 95)(36 100 40 96)(43 113 53 103)(44 114 54 104)(45 109 49 105)(46 110 50 106)(47 111 51 107)(48 112 52 108)(55 125 65 115)(56 126 66 116)(57 121 61 117)(58 122 62 118)(59 123 63 119)(60 124 64 120)(127 187 137 197)(128 188 138 198)(129 189 133 193)(130 190 134 194)(131 191 135 195)(132 192 136 196)(139 199 149 209)(140 200 150 210)(141 201 145 205)(142 202 146 206)(143 203 147 207)(144 204 148 208)(151 211 161 221)(152 212 162 222)(153 213 157 217)(154 214 158 218)(155 215 159 219)(156 216 160 220)(163 223 173 233)(164 224 174 234)(165 225 169 229)(166 226 170 230)(167 227 171 231)(168 228 172 232)(175 235 185 246)(176 236 186 241)(177 237 181 242)(178 238 182 243)(179 239 183 244)(180 240 184 245)(247 310 258 315)(248 311 253 316)(249 312 254 317)(250 307 255 318)(251 308 256 313)(252 309 257 314)(259 322 270 327)(260 323 265 328)(261 324 266 329)(262 319 267 330)(263 320 268 325)(264 321 269 326)(271 334 282 339)(272 335 277 340)(273 336 278 341)(274 331 279 342)(275 332 280 337)(276 333 281 338)(283 346 294 351)(284 347 289 352)(285 348 290 353)(286 343 291 354)(287 344 292 349)(288 345 293 350)(295 358 306 363)(296 359 301 364)(297 360 302 365)(298 355 303 366)(299 356 304 361)(300 357 305 362)(367 435 378 430)(368 436 373 431)(369 437 374 432)(370 438 375 427)(371 433 376 428)(372 434 377 429)(379 447 390 442)(380 448 385 443)(381 449 386 444)(382 450 387 439)(383 445 388 440)(384 446 389 441)(391 459 402 454)(392 460 397 455)(393 461 398 456)(394 462 399 451)(395 457 400 452)(396 458 401 453)(403 471 414 466)(404 472 409 467)(405 473 410 468)(406 474 411 463)(407 469 412 464)(408 470 413 465)
(1 248 18 253)(2 247 13 258)(3 252 14 257)(4 251 15 256)(5 250 16 255)(6 249 17 254)(7 246 479 235)(8 245 480 240)(9 244 475 239)(10 243 476 238)(11 242 477 237)(12 241 478 236)(19 268 29 263)(20 267 30 262)(21 266 25 261)(22 265 26 260)(23 270 27 259)(24 269 28 264)(31 280 41 275)(32 279 42 274)(33 278 37 273)(34 277 38 272)(35 282 39 271)(36 281 40 276)(43 292 53 287)(44 291 54 286)(45 290 49 285)(46 289 50 284)(47 294 51 283)(48 293 52 288)(55 304 65 299)(56 303 66 298)(57 302 61 297)(58 301 62 296)(59 306 63 295)(60 305 64 300)(67 316 77 311)(68 315 78 310)(69 314 73 309)(70 313 74 308)(71 318 75 307)(72 317 76 312)(79 328 89 323)(80 327 90 322)(81 326 85 321)(82 325 86 320)(83 330 87 319)(84 329 88 324)(91 340 101 335)(92 339 102 334)(93 338 97 333)(94 337 98 332)(95 342 99 331)(96 341 100 336)(103 352 113 347)(104 351 114 346)(105 350 109 345)(106 349 110 344)(107 354 111 343)(108 353 112 348)(115 364 125 359)(116 363 126 358)(117 362 121 357)(118 361 122 356)(119 366 123 355)(120 365 124 360)(127 376 137 371)(128 375 138 370)(129 374 133 369)(130 373 134 368)(131 378 135 367)(132 377 136 372)(139 388 149 383)(140 387 150 382)(141 386 145 381)(142 385 146 380)(143 390 147 379)(144 389 148 384)(151 400 161 395)(152 399 162 394)(153 398 157 393)(154 397 158 392)(155 402 159 391)(156 401 160 396)(163 412 173 407)(164 411 174 406)(165 410 169 405)(166 409 170 404)(167 414 171 403)(168 413 172 408)(175 424 185 419)(176 423 186 418)(177 422 181 417)(178 421 182 416)(179 426 183 415)(180 425 184 420)(187 436 197 431)(188 435 198 430)(189 434 193 429)(190 433 194 428)(191 438 195 427)(192 437 196 432)(199 448 209 443)(200 447 210 442)(201 446 205 441)(202 445 206 440)(203 450 207 439)(204 449 208 444)(211 460 221 455)(212 459 222 454)(213 458 217 453)(214 457 218 452)(215 462 219 451)(216 461 220 456)(223 472 233 467)(224 471 234 466)(225 470 229 465)(226 469 230 464)(227 474 231 463)(228 473 232 468)
G:=sub<Sym(480)| (1,62,50,38,26)(2,63,51,39,27)(3,64,52,40,28)(4,65,53,41,29)(5,66,54,42,30)(6,61,49,37,25)(7,472,460,448,436)(8,473,461,449,437)(9,474,462,450,438)(10,469,457,445,433)(11,470,458,446,434)(12,471,459,447,435)(13,59,47,35,23)(14,60,48,36,24)(15,55,43,31,19)(16,56,44,32,20)(17,57,45,33,21)(18,58,46,34,22)(67,115,103,91,79)(68,116,104,92,80)(69,117,105,93,81)(70,118,106,94,82)(71,119,107,95,83)(72,120,108,96,84)(73,121,109,97,85)(74,122,110,98,86)(75,123,111,99,87)(76,124,112,100,88)(77,125,113,101,89)(78,126,114,102,90)(127,175,163,151,139)(128,176,164,152,140)(129,177,165,153,141)(130,178,166,154,142)(131,179,167,155,143)(132,180,168,156,144)(133,181,169,157,145)(134,182,170,158,146)(135,183,171,159,147)(136,184,172,160,148)(137,185,173,161,149)(138,186,174,162,150)(187,235,223,211,199)(188,236,224,212,200)(189,237,225,213,201)(190,238,226,214,202)(191,239,227,215,203)(192,240,228,216,204)(193,242,229,217,205)(194,243,230,218,206)(195,244,231,219,207)(196,245,232,220,208)(197,246,233,221,209)(198,241,234,222,210)(247,295,283,271,259)(248,296,284,272,260)(249,297,285,273,261)(250,298,286,274,262)(251,299,287,275,263)(252,300,288,276,264)(253,301,289,277,265)(254,302,290,278,266)(255,303,291,279,267)(256,304,292,280,268)(257,305,293,281,269)(258,306,294,282,270)(307,355,343,331,319)(308,356,344,332,320)(309,357,345,333,321)(310,358,346,334,322)(311,359,347,335,323)(312,360,348,336,324)(313,361,349,337,325)(314,362,350,338,326)(315,363,351,339,327)(316,364,352,340,328)(317,365,353,341,329)(318,366,354,342,330)(367,415,403,391,379)(368,416,404,392,380)(369,417,405,393,381)(370,418,406,394,382)(371,419,407,395,383)(372,420,408,396,384)(373,421,409,397,385)(374,422,410,398,386)(375,423,411,399,387)(376,424,412,400,388)(377,425,413,401,389)(378,426,414,402,390)(427,475,463,451,439)(428,476,464,452,440)(429,477,465,453,441)(430,478,466,454,442)(431,479,467,455,443)(432,480,468,456,444), (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), (1,371,4,368)(2,370,5,367)(3,369,6,372)(7,115,10,118)(8,120,11,117)(9,119,12,116)(13,375,16,378)(14,374,17,377)(15,373,18,376)(19,385,22,388)(20,390,23,387)(21,389,24,386)(25,384,28,381)(26,383,29,380)(27,382,30,379)(31,397,34,400)(32,402,35,399)(33,401,36,398)(37,396,40,393)(38,395,41,392)(39,394,42,391)(43,409,46,412)(44,414,47,411)(45,413,48,410)(49,408,52,405)(50,407,53,404)(51,406,54,403)(55,421,58,424)(56,426,59,423)(57,425,60,422)(61,420,64,417)(62,419,65,416)(63,418,66,415)(67,433,70,436)(68,438,71,435)(69,437,72,434)(73,432,76,429)(74,431,77,428)(75,430,78,427)(79,445,82,448)(80,450,83,447)(81,449,84,446)(85,444,88,441)(86,443,89,440)(87,442,90,439)(91,457,94,460)(92,462,95,459)(93,461,96,458)(97,456,100,453)(98,455,101,452)(99,454,102,451)(103,469,106,472)(104,474,107,471)(105,473,108,470)(109,468,112,465)(110,467,113,464)(111,466,114,463)(121,480,124,477)(122,479,125,476)(123,478,126,475)(127,251,130,248)(128,250,131,247)(129,249,132,252)(133,254,136,257)(134,253,137,256)(135,258,138,255)(139,263,142,260)(140,262,143,259)(141,261,144,264)(145,266,148,269)(146,265,149,268)(147,270,150,267)(151,275,154,272)(152,274,155,271)(153,273,156,276)(157,278,160,281)(158,277,161,280)(159,282,162,279)(163,287,166,284)(164,286,167,283)(165,285,168,288)(169,290,172,293)(170,289,173,292)(171,294,174,291)(175,299,178,296)(176,298,179,295)(177,297,180,300)(181,302,184,305)(182,301,185,304)(183,306,186,303)(187,311,190,308)(188,310,191,307)(189,309,192,312)(193,314,196,317)(194,313,197,316)(195,318,198,315)(199,323,202,320)(200,322,203,319)(201,321,204,324)(205,326,208,329)(206,325,209,328)(207,330,210,327)(211,335,214,332)(212,334,215,331)(213,333,216,336)(217,338,220,341)(218,337,221,340)(219,342,222,339)(223,347,226,344)(224,346,227,343)(225,345,228,348)(229,350,232,353)(230,349,233,352)(231,354,234,351)(235,359,238,356)(236,358,239,355)(237,357,240,360)(241,363,244,366)(242,362,245,365)(243,361,246,364), (1,70,18,74)(2,71,13,75)(3,72,14,76)(4,67,15,77)(5,68,16,78)(6,69,17,73)(7,421,479,416)(8,422,480,417)(9,423,475,418)(10,424,476,419)(11,425,477,420)(12,426,478,415)(19,89,29,79)(20,90,30,80)(21,85,25,81)(22,86,26,82)(23,87,27,83)(24,88,28,84)(31,101,41,91)(32,102,42,92)(33,97,37,93)(34,98,38,94)(35,99,39,95)(36,100,40,96)(43,113,53,103)(44,114,54,104)(45,109,49,105)(46,110,50,106)(47,111,51,107)(48,112,52,108)(55,125,65,115)(56,126,66,116)(57,121,61,117)(58,122,62,118)(59,123,63,119)(60,124,64,120)(127,187,137,197)(128,188,138,198)(129,189,133,193)(130,190,134,194)(131,191,135,195)(132,192,136,196)(139,199,149,209)(140,200,150,210)(141,201,145,205)(142,202,146,206)(143,203,147,207)(144,204,148,208)(151,211,161,221)(152,212,162,222)(153,213,157,217)(154,214,158,218)(155,215,159,219)(156,216,160,220)(163,223,173,233)(164,224,174,234)(165,225,169,229)(166,226,170,230)(167,227,171,231)(168,228,172,232)(175,235,185,246)(176,236,186,241)(177,237,181,242)(178,238,182,243)(179,239,183,244)(180,240,184,245)(247,310,258,315)(248,311,253,316)(249,312,254,317)(250,307,255,318)(251,308,256,313)(252,309,257,314)(259,322,270,327)(260,323,265,328)(261,324,266,329)(262,319,267,330)(263,320,268,325)(264,321,269,326)(271,334,282,339)(272,335,277,340)(273,336,278,341)(274,331,279,342)(275,332,280,337)(276,333,281,338)(283,346,294,351)(284,347,289,352)(285,348,290,353)(286,343,291,354)(287,344,292,349)(288,345,293,350)(295,358,306,363)(296,359,301,364)(297,360,302,365)(298,355,303,366)(299,356,304,361)(300,357,305,362)(367,435,378,430)(368,436,373,431)(369,437,374,432)(370,438,375,427)(371,433,376,428)(372,434,377,429)(379,447,390,442)(380,448,385,443)(381,449,386,444)(382,450,387,439)(383,445,388,440)(384,446,389,441)(391,459,402,454)(392,460,397,455)(393,461,398,456)(394,462,399,451)(395,457,400,452)(396,458,401,453)(403,471,414,466)(404,472,409,467)(405,473,410,468)(406,474,411,463)(407,469,412,464)(408,470,413,465), 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G:=Group( 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(1,248,18,253)(2,247,13,258)(3,252,14,257)(4,251,15,256)(5,250,16,255)(6,249,17,254)(7,246,479,235)(8,245,480,240)(9,244,475,239)(10,243,476,238)(11,242,477,237)(12,241,478,236)(19,268,29,263)(20,267,30,262)(21,266,25,261)(22,265,26,260)(23,270,27,259)(24,269,28,264)(31,280,41,275)(32,279,42,274)(33,278,37,273)(34,277,38,272)(35,282,39,271)(36,281,40,276)(43,292,53,287)(44,291,54,286)(45,290,49,285)(46,289,50,284)(47,294,51,283)(48,293,52,288)(55,304,65,299)(56,303,66,298)(57,302,61,297)(58,301,62,296)(59,306,63,295)(60,305,64,300)(67,316,77,311)(68,315,78,310)(69,314,73,309)(70,313,74,308)(71,318,75,307)(72,317,76,312)(79,328,89,323)(80,327,90,322)(81,326,85,321)(82,325,86,320)(83,330,87,319)(84,329,88,324)(91,340,101,335)(92,339,102,334)(93,338,97,333)(94,337,98,332)(95,342,99,331)(96,341,100,336)(103,352,113,347)(104,351,114,346)(105,350,109,345)(106,349,110,344)(107,354,111,343)(108,353,112,348)(115,364,125,359)(116,363,126,358)(117,362,121,357)(118,361,122,356)(119,366,123,355)(120,365,124,360)(127,376,137,371)(128,375,138,370)(129,374,133,369)(130,373,134,368)(131,378,135,367)(132,377,136,372)(139,388,149,383)(140,387,150,382)(141,386,145,381)(142,385,146,380)(143,390,147,379)(144,389,148,384)(151,400,161,395)(152,399,162,394)(153,398,157,393)(154,397,158,392)(155,402,159,391)(156,401,160,396)(163,412,173,407)(164,411,174,406)(165,410,169,405)(166,409,170,404)(167,414,171,403)(168,413,172,408)(175,424,185,419)(176,423,186,418)(177,422,181,417)(178,421,182,416)(179,426,183,415)(180,425,184,420)(187,436,197,431)(188,435,198,430)(189,434,193,429)(190,433,194,428)(191,438,195,427)(192,437,196,432)(199,448,209,443)(200,447,210,442)(201,446,205,441)(202,445,206,440)(203,450,207,439)(204,449,208,444)(211,460,221,455)(212,459,222,454)(213,458,217,453)(214,457,218,452)(215,462,219,451)(216,461,220,456)(223,472,233,467)(224,471,234,466)(225,470,229,465)(226,469,230,464)(227,474,231,463)(228,473,232,468) );
G=PermutationGroup([[(1,62,50,38,26),(2,63,51,39,27),(3,64,52,40,28),(4,65,53,41,29),(5,66,54,42,30),(6,61,49,37,25),(7,472,460,448,436),(8,473,461,449,437),(9,474,462,450,438),(10,469,457,445,433),(11,470,458,446,434),(12,471,459,447,435),(13,59,47,35,23),(14,60,48,36,24),(15,55,43,31,19),(16,56,44,32,20),(17,57,45,33,21),(18,58,46,34,22),(67,115,103,91,79),(68,116,104,92,80),(69,117,105,93,81),(70,118,106,94,82),(71,119,107,95,83),(72,120,108,96,84),(73,121,109,97,85),(74,122,110,98,86),(75,123,111,99,87),(76,124,112,100,88),(77,125,113,101,89),(78,126,114,102,90),(127,175,163,151,139),(128,176,164,152,140),(129,177,165,153,141),(130,178,166,154,142),(131,179,167,155,143),(132,180,168,156,144),(133,181,169,157,145),(134,182,170,158,146),(135,183,171,159,147),(136,184,172,160,148),(137,185,173,161,149),(138,186,174,162,150),(187,235,223,211,199),(188,236,224,212,200),(189,237,225,213,201),(190,238,226,214,202),(191,239,227,215,203),(192,240,228,216,204),(193,242,229,217,205),(194,243,230,218,206),(195,244,231,219,207),(196,245,232,220,208),(197,246,233,221,209),(198,241,234,222,210),(247,295,283,271,259),(248,296,284,272,260),(249,297,285,273,261),(250,298,286,274,262),(251,299,287,275,263),(252,300,288,276,264),(253,301,289,277,265),(254,302,290,278,266),(255,303,291,279,267),(256,304,292,280,268),(257,305,293,281,269),(258,306,294,282,270),(307,355,343,331,319),(308,356,344,332,320),(309,357,345,333,321),(310,358,346,334,322),(311,359,347,335,323),(312,360,348,336,324),(313,361,349,337,325),(314,362,350,338,326),(315,363,351,339,327),(316,364,352,340,328),(317,365,353,341,329),(318,366,354,342,330),(367,415,403,391,379),(368,416,404,392,380),(369,417,405,393,381),(370,418,406,394,382),(371,419,407,395,383),(372,420,408,396,384),(373,421,409,397,385),(374,422,410,398,386),(375,423,411,399,387),(376,424,412,400,388),(377,425,413,401,389),(378,426,414,402,390),(427,475,463,451,439),(428,476,464,452,440),(429,477,465,453,441),(430,478,466,454,442),(431,479,467,455,443),(432,480,468,456,444)], [(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)], [(1,371,4,368),(2,370,5,367),(3,369,6,372),(7,115,10,118),(8,120,11,117),(9,119,12,116),(13,375,16,378),(14,374,17,377),(15,373,18,376),(19,385,22,388),(20,390,23,387),(21,389,24,386),(25,384,28,381),(26,383,29,380),(27,382,30,379),(31,397,34,400),(32,402,35,399),(33,401,36,398),(37,396,40,393),(38,395,41,392),(39,394,42,391),(43,409,46,412),(44,414,47,411),(45,413,48,410),(49,408,52,405),(50,407,53,404),(51,406,54,403),(55,421,58,424),(56,426,59,423),(57,425,60,422),(61,420,64,417),(62,419,65,416),(63,418,66,415),(67,433,70,436),(68,438,71,435),(69,437,72,434),(73,432,76,429),(74,431,77,428),(75,430,78,427),(79,445,82,448),(80,450,83,447),(81,449,84,446),(85,444,88,441),(86,443,89,440),(87,442,90,439),(91,457,94,460),(92,462,95,459),(93,461,96,458),(97,456,100,453),(98,455,101,452),(99,454,102,451),(103,469,106,472),(104,474,107,471),(105,473,108,470),(109,468,112,465),(110,467,113,464),(111,466,114,463),(121,480,124,477),(122,479,125,476),(123,478,126,475),(127,251,130,248),(128,250,131,247),(129,249,132,252),(133,254,136,257),(134,253,137,256),(135,258,138,255),(139,263,142,260),(140,262,143,259),(141,261,144,264),(145,266,148,269),(146,265,149,268),(147,270,150,267),(151,275,154,272),(152,274,155,271),(153,273,156,276),(157,278,160,281),(158,277,161,280),(159,282,162,279),(163,287,166,284),(164,286,167,283),(165,285,168,288),(169,290,172,293),(170,289,173,292),(171,294,174,291),(175,299,178,296),(176,298,179,295),(177,297,180,300),(181,302,184,305),(182,301,185,304),(183,306,186,303),(187,311,190,308),(188,310,191,307),(189,309,192,312),(193,314,196,317),(194,313,197,316),(195,318,198,315),(199,323,202,320),(200,322,203,319),(201,321,204,324),(205,326,208,329),(206,325,209,328),(207,330,210,327),(211,335,214,332),(212,334,215,331),(213,333,216,336),(217,338,220,341),(218,337,221,340),(219,342,222,339),(223,347,226,344),(224,346,227,343),(225,345,228,348),(229,350,232,353),(230,349,233,352),(231,354,234,351),(235,359,238,356),(236,358,239,355),(237,357,240,360),(241,363,244,366),(242,362,245,365),(243,361,246,364)], [(1,70,18,74),(2,71,13,75),(3,72,14,76),(4,67,15,77),(5,68,16,78),(6,69,17,73),(7,421,479,416),(8,422,480,417),(9,423,475,418),(10,424,476,419),(11,425,477,420),(12,426,478,415),(19,89,29,79),(20,90,30,80),(21,85,25,81),(22,86,26,82),(23,87,27,83),(24,88,28,84),(31,101,41,91),(32,102,42,92),(33,97,37,93),(34,98,38,94),(35,99,39,95),(36,100,40,96),(43,113,53,103),(44,114,54,104),(45,109,49,105),(46,110,50,106),(47,111,51,107),(48,112,52,108),(55,125,65,115),(56,126,66,116),(57,121,61,117),(58,122,62,118),(59,123,63,119),(60,124,64,120),(127,187,137,197),(128,188,138,198),(129,189,133,193),(130,190,134,194),(131,191,135,195),(132,192,136,196),(139,199,149,209),(140,200,150,210),(141,201,145,205),(142,202,146,206),(143,203,147,207),(144,204,148,208),(151,211,161,221),(152,212,162,222),(153,213,157,217),(154,214,158,218),(155,215,159,219),(156,216,160,220),(163,223,173,233),(164,224,174,234),(165,225,169,229),(166,226,170,230),(167,227,171,231),(168,228,172,232),(175,235,185,246),(176,236,186,241),(177,237,181,242),(178,238,182,243),(179,239,183,244),(180,240,184,245),(247,310,258,315),(248,311,253,316),(249,312,254,317),(250,307,255,318),(251,308,256,313),(252,309,257,314),(259,322,270,327),(260,323,265,328),(261,324,266,329),(262,319,267,330),(263,320,268,325),(264,321,269,326),(271,334,282,339),(272,335,277,340),(273,336,278,341),(274,331,279,342),(275,332,280,337),(276,333,281,338),(283,346,294,351),(284,347,289,352),(285,348,290,353),(286,343,291,354),(287,344,292,349),(288,345,293,350),(295,358,306,363),(296,359,301,364),(297,360,302,365),(298,355,303,366),(299,356,304,361),(300,357,305,362),(367,435,378,430),(368,436,373,431),(369,437,374,432),(370,438,375,427),(371,433,376,428),(372,434,377,429),(379,447,390,442),(380,448,385,443),(381,449,386,444),(382,450,387,439),(383,445,388,440),(384,446,389,441),(391,459,402,454),(392,460,397,455),(393,461,398,456),(394,462,399,451),(395,457,400,452),(396,458,401,453),(403,471,414,466),(404,472,409,467),(405,473,410,468),(406,474,411,463),(407,469,412,464),(408,470,413,465)], [(1,248,18,253),(2,247,13,258),(3,252,14,257),(4,251,15,256),(5,250,16,255),(6,249,17,254),(7,246,479,235),(8,245,480,240),(9,244,475,239),(10,243,476,238),(11,242,477,237),(12,241,478,236),(19,268,29,263),(20,267,30,262),(21,266,25,261),(22,265,26,260),(23,270,27,259),(24,269,28,264),(31,280,41,275),(32,279,42,274),(33,278,37,273),(34,277,38,272),(35,282,39,271),(36,281,40,276),(43,292,53,287),(44,291,54,286),(45,290,49,285),(46,289,50,284),(47,294,51,283),(48,293,52,288),(55,304,65,299),(56,303,66,298),(57,302,61,297),(58,301,62,296),(59,306,63,295),(60,305,64,300),(67,316,77,311),(68,315,78,310),(69,314,73,309),(70,313,74,308),(71,318,75,307),(72,317,76,312),(79,328,89,323),(80,327,90,322),(81,326,85,321),(82,325,86,320),(83,330,87,319),(84,329,88,324),(91,340,101,335),(92,339,102,334),(93,338,97,333),(94,337,98,332),(95,342,99,331),(96,341,100,336),(103,352,113,347),(104,351,114,346),(105,350,109,345),(106,349,110,344),(107,354,111,343),(108,353,112,348),(115,364,125,359),(116,363,126,358),(117,362,121,357),(118,361,122,356),(119,366,123,355),(120,365,124,360),(127,376,137,371),(128,375,138,370),(129,374,133,369),(130,373,134,368),(131,378,135,367),(132,377,136,372),(139,388,149,383),(140,387,150,382),(141,386,145,381),(142,385,146,380),(143,390,147,379),(144,389,148,384),(151,400,161,395),(152,399,162,394),(153,398,157,393),(154,397,158,392),(155,402,159,391),(156,401,160,396),(163,412,173,407),(164,411,174,406),(165,410,169,405),(166,409,170,404),(167,414,171,403),(168,413,172,408),(175,424,185,419),(176,423,186,418),(177,422,181,417),(178,421,182,416),(179,426,183,415),(180,425,184,420),(187,436,197,431),(188,435,198,430),(189,434,193,429),(190,433,194,428),(191,438,195,427),(192,437,196,432),(199,448,209,443),(200,447,210,442),(201,446,205,441),(202,445,206,440),(203,450,207,439),(204,449,208,444),(211,460,221,455),(212,459,222,454),(213,458,217,453),(214,457,218,452),(215,462,219,451),(216,461,220,456),(223,472,233,467),(224,471,234,466),(225,470,229,465),(226,469,230,464),(227,474,231,463),(228,473,232,468)]])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | ··· | 10L | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20P | 20Q | ··· | 20AF | 20AG | ··· | 20AN | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | S3 | Q8 | D6 | C4○D4 | C5×S3 | C5×Q8 | C4○D12 | S3×C10 | C5×C4○D4 | C5×C4○D12 | D4⋊2S3 | S3×Q8 | C5×D4⋊2S3 | C5×S3×Q8 |
| kernel | C5×Dic3.Q8 | Dic3×C20 | C5×Dic3⋊C4 | C5×C4⋊Dic3 | C15×C4⋊C4 | Dic3.Q8 | C4×Dic3 | Dic3⋊C4 | C4⋊Dic3 | C3×C4⋊C4 | C5×C4⋊C4 | C5×Dic3 | C2×C20 | C30 | C4⋊C4 | Dic3 | C10 | C2×C4 | C6 | C2 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 16 | 4 | 4 | 1 | 2 | 3 | 4 | 4 | 8 | 4 | 12 | 16 | 16 | 1 | 1 | 4 | 4 |
Matrix representation of C5×Dic3.Q8 ►in GL4(𝔽61) generated by
| 58 | 0 | 0 | 0 |
| 0 | 58 | 0 | 0 |
| 0 | 0 | 58 | 0 |
| 0 | 0 | 0 | 58 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 1 |
| 30 | 40 | 0 | 0 |
| 40 | 31 | 0 | 0 |
| 0 | 0 | 34 | 31 |
| 0 | 0 | 4 | 27 |
| 0 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 52 | 18 |
| 0 | 0 | 43 | 9 |
| 50 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 |
| 0 | 0 | 53 | 25 |
| 0 | 0 | 17 | 8 |
G:=sub<GL(4,GF(61))| [58,0,0,0,0,58,0,0,0,0,58,0,0,0,0,58],[60,0,0,0,0,60,0,0,0,0,0,60,0,0,1,1],[30,40,0,0,40,31,0,0,0,0,34,4,0,0,31,27],[0,60,0,0,1,0,0,0,0,0,52,43,0,0,18,9],[50,0,0,0,0,50,0,0,0,0,53,17,0,0,25,8] >;
C5×Dic3.Q8 in GAP, Magma, Sage, TeX
C_5\times {\rm Dic}_3.Q_8 % in TeX
G:=Group("C5xDic3.Q8"); // GroupNames label
G:=SmallGroup(480,768);
// by ID
G=gap.SmallGroup(480,768);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,288,926,891,436,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^6=d^4=1,c^2=b^3,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c,c*e=e*c,e*d*e^-1=b^3*d^-1>;
// generators/relations