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G = C5×Q8×Dic3  order 480 = 25·3·5

Direct product of C5, Q8 and Dic3

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

Aliases: C5×Q8×Dic3, C3⋊3(Q8×C20), C15⋊23(C4×Q8), (C3×Q8)⋊3C20, (Q8×C15)⋊15C4, (C6×Q8).5C10, C10.54(S3×Q8), C6.16(Q8×C10), C60.184(C2×C4), C12.14(C2×C20), (C2×C20).366D6, (Q8×C30).15C2, (Q8×C10).14S3, C30.114(C2×Q8), C4.4(C10×Dic3), C4⋊Dic3.12C10, C6.26(C22×C20), (C4×Dic3).5C10, C20.54(C2×Dic3), C30.271(C4○D4), (C2×C60).367C22, C30.233(C22×C4), (C2×C30).436C23, (Dic3×C20).14C2, C10.53(Q8⋊3S3), C10.49(C22×Dic3), (C10×Dic3).232C22, C2.3(C5×S3×Q8), C6.35(C5×C4○D4), (C2×Q8).7(C5×S3), C2.7(Dic3×C2×C10), (C2×C4).56(S3×C10), C22.26(S3×C2×C10), C2.3(C5×Q8⋊3S3), (C2×C12).40(C2×C10), (C5×C4⋊Dic3).26C2, (C2×C6).57(C22×C10), (C2×C10).370(C22×S3), (C2×Dic3).41(C2×C10), SmallGroup(480,824)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C5×Q8×Dic3
Chief series
C1 — C3 — C6 — C2×C6 — C2×C30 — C10×Dic3 — Dic3×C20 — C5×Q8×Dic3
Lower central
C3 — C6 — C5×Q8×Dic3
Upper central
C1 — C2×C10 — Q8×C10

Generators and relations for C5×Q8×Dic3
 G = < a,b,c,d,e | a5=b4=d6=1, c2=b2, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 228 in 140 conjugacy classes, 102 normal (28 characteristic)
C1, C2, C3, C4, C4, C22, C5, C6, C2×C4, C2×C4, Q8, C10, Dic3, Dic3, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, C20, C20, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C3×Q8, C30, C4×Q8, C2×C20, C2×C20, C5×Q8, C4×Dic3, C4⋊Dic3, C6×Q8, C5×Dic3, C5×Dic3, C60, C2×C30, C4×C20, C5×C4⋊C4, Q8×C10, Q8×Dic3, C10×Dic3, C10×Dic3, C2×C60, Q8×C15, Q8×C20, Dic3×C20, C5×C4⋊Dic3, Q8×C30, C5×Q8×Dic3
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, Q8, C23, C10, Dic3, D6, C22×C4, C2×Q8, C4○D4, C20, C2×C10, C2×Dic3, C22×S3, C5×S3, C4×Q8, C2×C20, C5×Q8, C22×C10, S3×Q8, Q8⋊3S3, C22×Dic3, C5×Dic3, S3×C10, C22×C20, Q8×C10, C5×C4○D4, Q8×Dic3, C10×Dic3, S3×C2×C10, Q8×C20, C5×S3×Q8, C5×Q8⋊3S3, Dic3×C2×C10, C5×Q8×Dic3

Smallest permutation representation of C5×Q8×Dic3
►Regular action on 480 points
Generators in S480
(1 60 48 36 24)(2 55 43 31 19)(3 56 44 32 20)(4 57 45 33 21)(5 58 46 34 22)(6 59 47 35 23)(7 470 458 446 434)(8 471 459 447 435)(9 472 460 448 436)(10 473 461 449 437)(11 474 462 450 438)(12 469 457 445 433)(13 61 49 37 25)(14 62 50 38 26)(15 63 51 39 27)(16 64 52 40 28)(17 65 53 41 29)(18 66 54 42 30)(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)(193 246 229 217 205)(194 241 230 218 206)(195 242 231 219 207)(196 243 232 220 208)(197 244 233 221 209)(198 245 234 222 210)(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 76 16 72)(2 77 17 67)(3 78 18 68)(4 73 13 69)(5 74 14 70)(6 75 15 71)(7 417 477 422)(8 418 478 423)(9 419 479 424)(10 420 480 425)(11 415 475 426)(12 416 476 421)(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)(175 235 185 244)(176 236 186 245)(177 237 181 246)(178 238 182 241)(179 239 183 242)(180 240 184 243)(247 318 258 307)(248 313 253 308)(249 314 254 309)(250 315 255 310)(251 316 256 311)(252 317 257 312)(259 330 270 319)(260 325 265 320)(261 326 266 321)(262 327 267 322)(263 328 268 323)(264 329 269 324)(271 342 282 331)(272 337 277 332)(273 338 278 333)(274 339 279 334)(275 340 280 335)(276 341 281 336)(283 354 294 343)(284 349 289 344)(285 350 290 345)(286 351 291 346)(287 352 292 347)(288 353 293 348)(295 366 306 355)(296 361 301 356)(297 362 302 357)(298 363 303 358)(299 364 304 359)(300 365 305 360)(367 427 378 438)(368 428 373 433)(369 429 374 434)(370 430 375 435)(371 431 376 436)(372 432 377 437)(379 439 390 450)(380 440 385 445)(381 441 386 446)(382 442 387 447)(383 443 388 448)(384 444 389 449)(391 451 402 462)(392 452 397 457)(393 453 398 458)(394 454 399 459)(395 455 400 460)(396 456 401 461)(403 463 414 474)(404 464 409 469)(405 465 410 470)(406 466 411 471)(407 467 412 472)(408 468 413 473)

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

(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)

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

150 conjugacy classes

class 1 2A2B2C 3 4A···4F4G4H4I4J4K···4P5A5B5C5D6A6B6C10A···10L12A···12F15A15B15C15D20A···20X20Y···20AN20AO···20BL30A···30L60A···60X
order122234···444444···4555566610···1012···121515151520···2020···2020···2030···3060···60
size111122···233336···611112221···14···422222···23···36···62···24···4

150 irreducible representations

dim111111111122222222224444
type+++++-+--+
imageC1C2C2C2C4C5C10C10C10C20S3Q8D6Dic3C4○D4C5×S3C5×Q8S3×C10C5×Dic3C5×C4○D4S3×Q8Q8⋊3S3C5×S3×Q8C5×Q8⋊3S3
kernelC5×Q8×Dic3Dic3×C20C5×C4⋊Dic3Q8×C30Q8×C15Q8×Dic3C4×Dic3C4⋊Dic3C6×Q8C3×Q8Q8×C10C5×Dic3C2×C20C5×Q8C30C2×Q8Dic3C2×C4Q8C6C10C10C2C2
# reps13318412124321234248121681144

Matrix representation of C5×Q8×Dic3 ►in GL5(𝔽61)

10000
01000
00100
000340
000034
,
600000
0605900
01100
000600
000060
,
10000
0173800
024400
00010
00001
,
600000
01000
00100
000601
000600
,
500000
01000
00100
0003751
0002724

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,34,0,0,0,0,0,34],[60,0,0,0,0,0,60,1,0,0,0,59,1,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,17,2,0,0,0,38,44,0,0,0,0,0,1,0,0,0,0,0,1],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,60,0,0,0,1,0],[50,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,37,27,0,0,0,51,24] >;
 

C5×Q8×Dic3 in GAP, Magma, Sage, TeX 

C_5\times Q_8\times {\rm Dic}_3
 
% in TeX
 
G:=Group("C5xQ8xDic3");
 
// GroupNames label
 
G:=SmallGroup(480,824);
 
// by ID
 
G=gap.SmallGroup(480,824);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,280,568,891,436,15686]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^5=b^4=d^6=1,c^2=b^2,e^2=d^3,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,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

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