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G = C5×Dic3⋊Q8order 480 = 25·3·5

Direct product of C5 and Dic3⋊Q8

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

Aliases: C5×Dic3⋊Q8, C60.154D4, C1518(C4⋊Q8), (C5×Dic3)⋊9Q8, Dic32(C5×Q8), C12.21(C5×D4), C6.56(D4×C10), (C6×Q8).4C10, C10.53(S3×Q8), C6.15(Q8×C10), (C2×C20).246D6, C30.439(C2×D4), (Q8×C30).14C2, (Q8×C10).13S3, C30.113(C2×Q8), C20.75(C3⋊D4), Dic3⋊C4.6C10, (C4×Dic3).4C10, (C2×Dic6).9C10, (C2×C60).430C22, (C2×C30).435C23, (C10×Dic6).19C2, (Dic3×C20).13C2, (C10×Dic3).151C22, C33(C5×C4⋊Q8), C2.8(C5×S3×Q8), C4.10(C5×C3⋊D4), (C2×Q8).6(C5×S3), (C2×C4).55(S3×C10), C2.20(C10×C3⋊D4), C22.63(S3×C2×C10), (C2×C12).64(C2×C10), C10.141(C2×C3⋊D4), (C5×Dic3⋊C4).16C2, (C2×C6).56(C22×C10), (C2×C10).369(C22×S3), (C2×Dic3).40(C2×C10), SmallGroup(480,823)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C5×Dic3⋊Q8
C1C3C6C2×C6C2×C30C10×Dic3Dic3×C20 — C5×Dic3⋊Q8
C3C2×C6 — C5×Dic3⋊Q8
C1C2×C10Q8×C10

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=b-1, bd=db, be=eb, dcd-1=b3c, ce=ec, ede-1=d-1 >

Subgroups: 260 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], Dic3 [×4], Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C15, C42, C4⋊C4 [×4], C2×Q8, C2×Q8, C20 [×2], C20 [×8], C2×C10, Dic6 [×2], C2×Dic3 [×4], C2×C12, C2×C12 [×2], C3×Q8 [×2], C30, C30 [×2], C4⋊Q8, C2×C20, C2×C20 [×2], C2×C20 [×4], C5×Q8 [×4], C4×Dic3, Dic3⋊C4 [×4], C2×Dic6, C6×Q8, C5×Dic3 [×4], C5×Dic3 [×2], C60 [×2], C60 [×2], C2×C30, C4×C20, C5×C4⋊C4 [×4], Q8×C10, Q8×C10, Dic3⋊Q8, C5×Dic6 [×2], C10×Dic3 [×4], C2×C60, C2×C60 [×2], Q8×C15 [×2], C5×C4⋊Q8, Dic3×C20, C5×Dic3⋊C4 [×4], C10×Dic6, Q8×C30, C5×Dic3⋊Q8
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], Q8 [×4], C23, C10 [×7], D6 [×3], C2×D4, C2×Q8 [×2], C2×C10 [×7], C3⋊D4 [×2], C22×S3, C5×S3, C4⋊Q8, C5×D4 [×2], C5×Q8 [×4], C22×C10, S3×Q8 [×2], C2×C3⋊D4, S3×C10 [×3], D4×C10, Q8×C10 [×2], Dic3⋊Q8, C5×C3⋊D4 [×2], S3×C2×C10, C5×C4⋊Q8, C5×S3×Q8 [×2], C10×C3⋊D4, C5×Dic3⋊Q8

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

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

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

120 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F4G4H4I4J5A5B5C5D6A6B6C10A···10L12A···12F15A15B15C15D20A···20H20I···20P20Q···20AF20AG···20AN30A···30L60A···60X
order122234444444444555566610···1012···121515151520···2020···2020···2020···2030···3060···60
size1111222446666121211112221···14···422222···24···46···612···122···24···4

120 irreducible representations

dim1111111111222222222244
type++++++-++-
imageC1C2C2C2C2C5C10C10C10C10S3Q8D4D6C3⋊D4C5×S3C5×Q8C5×D4S3×C10C5×C3⋊D4S3×Q8C5×S3×Q8
kernelC5×Dic3⋊Q8Dic3×C20C5×Dic3⋊C4C10×Dic6Q8×C30Dic3⋊Q8C4×Dic3Dic3⋊C4C2×Dic6C6×Q8Q8×C10C5×Dic3C60C2×C20C20C2×Q8Dic3C12C2×C4C4C10C2
# reps11411441644142344168121628

Matrix representation of C5×Dic3⋊Q8 in GL4(𝔽61) generated by

9000
0900
0010
0001
,
1100
60000
00600
00060
,
275700
303400
00940
00152
,
91800
435200
00500
00811
,
60000
06000
00940
00152
G:=sub<GL(4,GF(61))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[1,60,0,0,1,0,0,0,0,0,60,0,0,0,0,60],[27,30,0,0,57,34,0,0,0,0,9,1,0,0,40,52],[9,43,0,0,18,52,0,0,0,0,50,8,0,0,0,11],[60,0,0,0,0,60,0,0,0,0,9,1,0,0,40,52] >;

C5×Dic3⋊Q8 in GAP, Magma, Sage, TeX

C_5\times {\rm Dic}_3\rtimes Q_8
% in TeX

G:=Group("C5xDic3:Q8");
// GroupNames label

G:=SmallGroup(480,823);
// by ID

G=gap.SmallGroup(480,823);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,560,288,1766,471,226,15686]);
// 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=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^3*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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