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G = C5×Q82Dic3order 480 = 25·3·5

Direct product of C5 and Q82Dic3

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

Aliases: C5×Q82Dic3, C60.142D4, C30.24Q16, C30.39SD16, (C3×Q8)⋊1C20, C12.9(C5×D4), C6.5(C5×Q16), C12.8(C2×C20), (Q8×C15)⋊13C4, Q82(C5×Dic3), (C6×Q8).1C10, C6.8(C5×SD16), C60.178(C2×C4), (C5×Q8)⋊10Dic3, (C2×C30).177D4, (C2×C20).351D6, (Q8×C30).11C2, (Q8×C10).10S3, C4.2(C10×Dic3), C1519(Q8⋊C4), C20.93(C3⋊D4), C4⋊Dic3.10C10, C20.52(C2×Dic3), (C2×C60).347C22, C10.12(C3⋊Q16), C30.120(C22⋊C4), C10.12(Q82S3), C10.36(C6.D4), (C2×C3⋊C8).5C10, C33(C5×Q8⋊C4), (C10×C3⋊C8).17C2, (C2×C6).34(C5×D4), C4.14(C5×C3⋊D4), (C2×Q8).3(C5×S3), C2.3(C5×C3⋊Q16), (C2×C4).39(S3×C10), C6.16(C5×C22⋊C4), C2.3(C5×Q82S3), (C2×C12).17(C2×C10), (C5×C4⋊Dic3).24C2, C2.6(C5×C6.D4), C22.18(C5×C3⋊D4), (C2×C10).90(C3⋊D4), SmallGroup(480,154)

Series: Derived Chief Lower central Upper central

C1C12 — C5×Q82Dic3
C1C3C6C12C2×C12C2×C60C5×C4⋊Dic3 — C5×Q82Dic3
C3C6C12 — C5×Q82Dic3
C1C2×C10C2×C20Q8×C10

Generators and relations for C5×Q82Dic3
 G = < a,b,c,d,e | a5=b4=d6=1, c2=b2, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=d-1 >

Subgroups: 164 in 84 conjugacy classes, 50 normal (42 characteristic)
C1, C2 [×3], C3, C4 [×2], C4 [×3], C22, C5, C6 [×3], C8, C2×C4, C2×C4 [×2], Q8 [×2], Q8, C10 [×3], Dic3, C12 [×2], C12 [×2], C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, C20 [×2], C20 [×3], C2×C10, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C3×Q8 [×2], C3×Q8, C30 [×3], Q8⋊C4, C40, C2×C20, C2×C20 [×2], C5×Q8 [×2], C5×Q8, C2×C3⋊C8, C4⋊Dic3, C6×Q8, C5×Dic3, C60 [×2], C60 [×2], C2×C30, C5×C4⋊C4, C2×C40, Q8×C10, Q82Dic3, C5×C3⋊C8, C10×Dic3, C2×C60, C2×C60, Q8×C15 [×2], Q8×C15, C5×Q8⋊C4, C10×C3⋊C8, C5×C4⋊Dic3, Q8×C30, C5×Q82Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, C5, S3, C2×C4, D4 [×2], C10 [×3], Dic3 [×2], D6, C22⋊C4, SD16, Q16, C20 [×2], C2×C10, C2×Dic3, C3⋊D4 [×2], C5×S3, Q8⋊C4, C2×C20, C5×D4 [×2], Q82S3, C3⋊Q16, C6.D4, C5×Dic3 [×2], S3×C10, C5×C22⋊C4, C5×SD16, C5×Q16, Q82Dic3, C10×Dic3, C5×C3⋊D4 [×2], C5×Q8⋊C4, C5×Q82S3, C5×C3⋊Q16, C5×C6.D4, C5×Q82Dic3

Smallest permutation representation of C5×Q82Dic3
Regular action on 480 points
Generators in S480
(1 110 86 62 38)(2 111 87 63 39)(3 112 88 64 40)(4 113 89 65 41)(5 114 90 66 42)(6 109 85 61 37)(7 103 79 55 31)(8 104 80 56 32)(9 105 81 57 33)(10 106 82 58 34)(11 107 83 59 35)(12 108 84 60 36)(13 460 436 412 388)(14 461 437 413 389)(15 462 438 414 390)(16 457 433 409 385)(17 458 434 410 386)(18 459 435 411 387)(19 115 91 67 43)(20 116 92 68 44)(21 117 93 69 45)(22 118 94 70 46)(23 119 95 71 47)(24 120 96 72 48)(25 121 97 73 49)(26 122 98 74 50)(27 123 99 75 51)(28 124 100 76 52)(29 125 101 77 53)(30 126 102 78 54)(127 223 199 175 151)(128 224 200 176 152)(129 225 201 177 153)(130 226 202 178 154)(131 227 203 179 155)(132 228 204 180 156)(133 229 205 181 157)(134 230 206 182 158)(135 231 207 183 159)(136 232 208 184 160)(137 233 209 185 161)(138 234 210 186 162)(139 235 211 187 163)(140 236 212 188 164)(141 237 213 189 165)(142 238 214 190 166)(143 239 215 191 167)(144 240 216 192 168)(145 242 217 193 169)(146 243 218 194 170)(147 244 219 195 171)(148 245 220 196 172)(149 246 221 197 173)(150 241 222 198 174)(247 343 319 295 271)(248 344 320 296 272)(249 345 321 297 273)(250 346 322 298 274)(251 347 323 299 275)(252 348 324 300 276)(253 349 325 301 277)(254 350 326 302 278)(255 351 327 303 279)(256 352 328 304 280)(257 353 329 305 281)(258 354 330 306 282)(259 355 331 307 283)(260 356 332 308 284)(261 357 333 309 285)(262 358 334 310 286)(263 359 335 311 287)(264 360 336 312 288)(265 361 337 313 289)(266 362 338 314 290)(267 363 339 315 291)(268 364 340 316 292)(269 365 341 317 293)(270 366 342 318 294)(367 463 439 415 391)(368 464 440 416 392)(369 465 441 417 393)(370 466 442 418 394)(371 467 443 419 395)(372 468 444 420 396)(373 469 445 421 397)(374 470 446 422 398)(375 471 447 423 399)(376 472 448 424 400)(377 473 449 425 401)(378 474 450 426 402)(379 475 451 427 403)(380 476 452 428 404)(381 477 453 429 405)(382 478 454 430 406)(383 479 455 431 407)(384 480 456 432 408)
(1 29 19 12)(2 30 20 7)(3 25 21 8)(4 26 22 9)(5 27 23 10)(6 28 24 11)(13 479 463 471)(14 480 464 472)(15 475 465 473)(16 476 466 474)(17 477 467 469)(18 478 468 470)(31 39 54 44)(32 40 49 45)(33 41 50 46)(34 42 51 47)(35 37 52 48)(36 38 53 43)(55 63 78 68)(56 64 73 69)(57 65 74 70)(58 66 75 71)(59 61 76 72)(60 62 77 67)(79 87 102 92)(80 88 97 93)(81 89 98 94)(82 90 99 95)(83 85 100 96)(84 86 101 91)(103 111 126 116)(104 112 121 117)(105 113 122 118)(106 114 123 119)(107 109 124 120)(108 110 125 115)(127 140 150 135)(128 141 145 136)(129 142 146 137)(130 143 147 138)(131 144 148 133)(132 139 149 134)(151 164 174 159)(152 165 169 160)(153 166 170 161)(154 167 171 162)(155 168 172 157)(156 163 173 158)(175 188 198 183)(176 189 193 184)(177 190 194 185)(178 191 195 186)(179 192 196 181)(180 187 197 182)(199 212 222 207)(200 213 217 208)(201 214 218 209)(202 215 219 210)(203 216 220 205)(204 211 221 206)(223 236 241 231)(224 237 242 232)(225 238 243 233)(226 239 244 234)(227 240 245 229)(228 235 246 230)(247 263 268 255)(248 264 269 256)(249 259 270 257)(250 260 265 258)(251 261 266 253)(252 262 267 254)(271 287 292 279)(272 288 293 280)(273 283 294 281)(274 284 289 282)(275 285 290 277)(276 286 291 278)(295 311 316 303)(296 312 317 304)(297 307 318 305)(298 308 313 306)(299 309 314 301)(300 310 315 302)(319 335 340 327)(320 336 341 328)(321 331 342 329)(322 332 337 330)(323 333 338 325)(324 334 339 326)(343 359 364 351)(344 360 365 352)(345 355 366 353)(346 356 361 354)(347 357 362 349)(348 358 363 350)(367 375 388 383)(368 376 389 384)(369 377 390 379)(370 378 385 380)(371 373 386 381)(372 374 387 382)(391 399 412 407)(392 400 413 408)(393 401 414 403)(394 402 409 404)(395 397 410 405)(396 398 411 406)(415 423 436 431)(416 424 437 432)(417 425 438 427)(418 426 433 428)(419 421 434 429)(420 422 435 430)(439 447 460 455)(440 448 461 456)(441 449 462 451)(442 450 457 452)(443 445 458 453)(444 446 459 454)
(1 142 19 137)(2 143 20 138)(3 144 21 133)(4 139 22 134)(5 140 23 135)(6 141 24 136)(7 147 30 130)(8 148 25 131)(9 149 26 132)(10 150 27 127)(11 145 28 128)(12 146 29 129)(13 356 463 354)(14 357 464 349)(15 358 465 350)(16 359 466 351)(17 360 467 352)(18 355 468 353)(31 171 54 154)(32 172 49 155)(33 173 50 156)(34 174 51 151)(35 169 52 152)(36 170 53 153)(37 165 48 160)(38 166 43 161)(39 167 44 162)(40 168 45 157)(41 163 46 158)(42 164 47 159)(55 195 78 178)(56 196 73 179)(57 197 74 180)(58 198 75 175)(59 193 76 176)(60 194 77 177)(61 189 72 184)(62 190 67 185)(63 191 68 186)(64 192 69 181)(65 187 70 182)(66 188 71 183)(79 219 102 202)(80 220 97 203)(81 221 98 204)(82 222 99 199)(83 217 100 200)(84 218 101 201)(85 213 96 208)(86 214 91 209)(87 215 92 210)(88 216 93 205)(89 211 94 206)(90 212 95 207)(103 244 126 226)(104 245 121 227)(105 246 122 228)(106 241 123 223)(107 242 124 224)(108 243 125 225)(109 237 120 232)(110 238 115 233)(111 239 116 234)(112 240 117 229)(113 235 118 230)(114 236 119 231)(247 378 268 380)(248 373 269 381)(249 374 270 382)(250 375 265 383)(251 376 266 384)(252 377 267 379)(253 389 261 368)(254 390 262 369)(255 385 263 370)(256 386 264 371)(257 387 259 372)(258 388 260 367)(271 402 292 404)(272 397 293 405)(273 398 294 406)(274 399 289 407)(275 400 290 408)(276 401 291 403)(277 413 285 392)(278 414 286 393)(279 409 287 394)(280 410 288 395)(281 411 283 396)(282 412 284 391)(295 426 316 428)(296 421 317 429)(297 422 318 430)(298 423 313 431)(299 424 314 432)(300 425 315 427)(301 437 309 416)(302 438 310 417)(303 433 311 418)(304 434 312 419)(305 435 307 420)(306 436 308 415)(319 450 340 452)(320 445 341 453)(321 446 342 454)(322 447 337 455)(323 448 338 456)(324 449 339 451)(325 461 333 440)(326 462 334 441)(327 457 335 442)(328 458 336 443)(329 459 331 444)(330 460 332 439)(343 474 364 476)(344 469 365 477)(345 470 366 478)(346 471 361 479)(347 472 362 480)(348 473 363 475)
(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)
(1 258 4 255)(2 257 5 254)(3 256 6 253)(7 249 10 252)(8 248 11 251)(9 247 12 250)(13 246 16 243)(14 245 17 242)(15 244 18 241)(19 260 22 263)(20 259 23 262)(21 264 24 261)(25 269 28 266)(26 268 29 265)(27 267 30 270)(31 273 34 276)(32 272 35 275)(33 271 36 274)(37 277 40 280)(38 282 41 279)(39 281 42 278)(43 284 46 287)(44 283 47 286)(45 288 48 285)(49 293 52 290)(50 292 53 289)(51 291 54 294)(55 297 58 300)(56 296 59 299)(57 295 60 298)(61 301 64 304)(62 306 65 303)(63 305 66 302)(67 308 70 311)(68 307 71 310)(69 312 72 309)(73 317 76 314)(74 316 77 313)(75 315 78 318)(79 321 82 324)(80 320 83 323)(81 319 84 322)(85 325 88 328)(86 330 89 327)(87 329 90 326)(91 332 94 335)(92 331 95 334)(93 336 96 333)(97 341 100 338)(98 340 101 337)(99 339 102 342)(103 345 106 348)(104 344 107 347)(105 343 108 346)(109 349 112 352)(110 354 113 351)(111 353 114 350)(115 356 118 359)(116 355 119 358)(117 360 120 357)(121 365 124 362)(122 364 125 361)(123 363 126 366)(127 369 130 372)(128 368 131 371)(129 367 132 370)(133 373 136 376)(134 378 137 375)(135 377 138 374)(139 380 142 383)(140 379 143 382)(141 384 144 381)(145 389 148 386)(146 388 149 385)(147 387 150 390)(151 393 154 396)(152 392 155 395)(153 391 156 394)(157 397 160 400)(158 402 161 399)(159 401 162 398)(163 404 166 407)(164 403 167 406)(165 408 168 405)(169 413 172 410)(170 412 173 409)(171 411 174 414)(175 417 178 420)(176 416 179 419)(177 415 180 418)(181 421 184 424)(182 426 185 423)(183 425 186 422)(187 428 190 431)(188 427 191 430)(189 432 192 429)(193 437 196 434)(194 436 197 433)(195 435 198 438)(199 441 202 444)(200 440 203 443)(201 439 204 442)(205 445 208 448)(206 450 209 447)(207 449 210 446)(211 452 214 455)(212 451 215 454)(213 456 216 453)(217 461 220 458)(218 460 221 457)(219 459 222 462)(223 465 226 468)(224 464 227 467)(225 463 228 466)(229 469 232 472)(230 474 233 471)(231 473 234 470)(235 476 238 479)(236 475 239 478)(237 480 240 477)

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

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

120 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E4F5A5B5C5D6A6B6C8A8B8C8D10A···10L12A···12F15A15B15C15D20A···20H20I···20P20Q···20X30A···30L40A···40P60A···60X
order122234444445555666888810···1012···121515151520···2020···2020···2030···3040···4060···60
size1111222441212111122266661···14···422222···24···412···122···26···64···4

120 irreducible representations

dim11111111112222222222222222224444
type++++++++--+-
imageC1C2C2C2C4C5C10C10C10C20S3D4D4D6Dic3SD16Q16C3⋊D4C3⋊D4C5×S3C5×D4C5×D4S3×C10C5×Dic3C5×SD16C5×Q16C5×C3⋊D4C5×C3⋊D4Q82S3C3⋊Q16C5×Q82S3C5×C3⋊Q16
kernelC5×Q82Dic3C10×C3⋊C8C5×C4⋊Dic3Q8×C30Q8×C15Q82Dic3C2×C3⋊C8C4⋊Dic3C6×Q8C3×Q8Q8×C10C60C2×C30C2×C20C5×Q8C30C30C20C2×C10C2×Q8C12C2×C6C2×C4Q8C6C6C4C22C10C10C2C2
# reps111144444161111222224444888881144

Matrix representation of C5×Q82Dic3 in GL6(𝔽241)

8700000
0870000
0098000
0009800
000010
000001
,
010000
24000000
000100
00240000
000010
000001
,
121780000
781200000
002119400
0019422000
000010
000001
,
100000
010000
00240000
00024000
00000240
00001240
,
1661470000
147750000
00452500
002519600
00007181
0000152170

G:=sub<GL(6,GF(241))| [87,0,0,0,0,0,0,87,0,0,0,0,0,0,98,0,0,0,0,0,0,98,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,240,0,0,0,0,1,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[121,78,0,0,0,0,78,120,0,0,0,0,0,0,21,194,0,0,0,0,194,220,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,0,0,0,0,240,240],[166,147,0,0,0,0,147,75,0,0,0,0,0,0,45,25,0,0,0,0,25,196,0,0,0,0,0,0,71,152,0,0,0,0,81,170] >;

C5×Q82Dic3 in GAP, Magma, Sage, TeX

C_5\times Q_8\rtimes_2{\rm Dic}_3
% in TeX

G:=Group("C5xQ8:2Dic3");
// GroupNames label

G:=SmallGroup(480,154);
// by ID

G=gap.SmallGroup(480,154);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,140,589,568,4204,2111,102,15686]);
// 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=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^-1>;
// generators/relations

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