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G = Dic3×C5⋊2C8  order 480 = 25·3·5

Direct product of Dic3 and C5⋊2C8

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

Aliases: Dic3×C5⋊2C8, C30.19C42, C15⋊9(C4×C8), C5⋊5(C8×Dic3), C15⋊3C8⋊14C4, C20.71(C4×S3), C10.22(S3×C8), C30.26(C2×C8), (C5×Dic3)⋊3C8, C12.75(C4×D5), C6.2(C4×Dic5), C60.142(C2×C4), (C2×C20).316D6, C4.18(D5×Dic3), (C2×C12).320D10, (Dic3×C20).6C2, (C4×Dic3).11D5, C10.14(C4×Dic3), C20.44(C2×Dic3), C2.2(Dic3×Dic5), C22.8(S3×Dic5), (C2×C60).218C22, C4.19(D30.C2), (C2×Dic3).7Dic5, (C10×Dic3).12C4, C3⋊1(C4×C5⋊2C8), (C3×C5⋊2C8)⋊3C4, C6.3(C2×C5⋊2C8), C2.2(S3×C5⋊2C8), (C6×C5⋊2C8).7C2, (C2×C5⋊2C8).13S3, (C2×C10).67(C4×S3), (C2×C30).72(C2×C4), (C2×C4).221(S3×D5), (C2×C6).9(C2×Dic5), (C2×C15⋊3C8).18C2, SmallGroup(480,26)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C15 — Dic3×C5⋊2C8
Chief series
C1 — C5 — C15 — C30 — C60 — C2×C60 — C6×C5⋊2C8 — Dic3×C5⋊2C8
Lower central
C15 — Dic3×C5⋊2C8
Upper central
C1 — C2×C4

Generators and relations for Dic3×C5⋊2C8
 G = < a,b,c,d | a6=c5=d8=1, b2=a3, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 220 in 88 conjugacy classes, 56 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, C10, C10, Dic3, C12, C2×C6, C15, C42, C2×C8, C20, C20, C2×C10, C3⋊C8, C24, C2×Dic3, C2×C12, C30, C30, C4×C8, C5⋊2C8, C5⋊2C8, C2×C20, C2×C20, C2×C3⋊C8, C4×Dic3, C2×C24, C5×Dic3, C60, C2×C30, C2×C5⋊2C8, C2×C5⋊2C8, C4×C20, C8×Dic3, C3×C5⋊2C8, C15⋊3C8, C10×Dic3, C2×C60, C4×C5⋊2C8, C6×C5⋊2C8, Dic3×C20, C2×C15⋊3C8, Dic3×C5⋊2C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D5, Dic3, D6, C42, C2×C8, Dic5, D10, C4×S3, C2×Dic3, C4×C8, C5⋊2C8, C4×D5, C2×Dic5, S3×C8, C4×Dic3, S3×D5, C2×C5⋊2C8, C4×Dic5, C8×Dic3, D5×Dic3, S3×Dic5, D30.C2, C4×C5⋊2C8, S3×C5⋊2C8, Dic3×Dic5, Dic3×C5⋊2C8

Smallest permutation representation of Dic3×C5⋊2C8
►Regular action on 480 points
Generators in S480
(1 445 25 306 104 175)(2 446 26 307 97 176)(3 447 27 308 98 169)(4 448 28 309 99 170)(5 441 29 310 100 171)(6 442 30 311 101 172)(7 443 31 312 102 173)(8 444 32 305 103 174)(9 246 228 365 51 190)(10 247 229 366 52 191)(11 248 230 367 53 192)(12 241 231 368 54 185)(13 242 232 361 55 186)(14 243 225 362 56 187)(15 244 226 363 49 188)(16 245 227 364 50 189)(17 279 84 433 155 463)(18 280 85 434 156 464)(19 273 86 435 157 457)(20 274 87 436 158 458)(21 275 88 437 159 459)(22 276 81 438 160 460)(23 277 82 439 153 461)(24 278 83 440 154 462)(33 261 47 390 452 412)(34 262 48 391 453 413)(35 263 41 392 454 414)(36 264 42 385 455 415)(37 257 43 386 456 416)(38 258 44 387 449 409)(39 259 45 388 450 410)(40 260 46 389 451 411)(57 418 354 380 268 320)(58 419 355 381 269 313)(59 420 356 382 270 314)(60 421 357 383 271 315)(61 422 358 384 272 316)(62 423 359 377 265 317)(63 424 360 378 266 318)(64 417 353 379 267 319)(65 329 215 224 194 291)(66 330 216 217 195 292)(67 331 209 218 196 293)(68 332 210 219 197 294)(69 333 211 220 198 295)(70 334 212 221 199 296)(71 335 213 222 200 289)(72 336 214 223 193 290)(73 477 139 147 91 179)(74 478 140 148 92 180)(75 479 141 149 93 181)(76 480 142 150 94 182)(77 473 143 151 95 183)(78 474 144 152 96 184)(79 475 137 145 89 177)(80 476 138 146 90 178)(105 349 236 401 303 131)(106 350 237 402 304 132)(107 351 238 403 297 133)(108 352 239 404 298 134)(109 345 240 405 299 135)(110 346 233 406 300 136)(111 347 234 407 301 129)(112 348 235 408 302 130)(113 255 470 166 428 128)(114 256 471 167 429 121)(115 249 472 168 430 122)(116 250 465 161 431 123)(117 251 466 162 432 124)(118 252 467 163 425 125)(119 253 468 164 426 126)(120 254 469 165 427 127)(201 400 340 327 288 370)(202 393 341 328 281 371)(203 394 342 321 282 372)(204 395 343 322 283 373)(205 396 344 323 284 374)(206 397 337 324 285 375)(207 398 338 325 286 376)(208 399 339 326 287 369)

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

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

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

96 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E···4L5A5B6A6B6C8A···8H8I···8P10A···10F12A12B12C12D15A15B20A···20H20I···20X24A···24H30A···30F60A···60H
order1222344444···4556668···88···810···1012121212151520···2020···2024···2430···3060···60
size1111211113···3222225···515···152···22222442···26···610···104···44···4

96 irreducible representations

dim111111112222222222244444
type++++++-+-++-+-
imageC1C2C2C2C4C4C4C8S3D5Dic3D6Dic5D10C4×S3C4×S3C5⋊2C8C4×D5S3×C8S3×D5D5×Dic3D30.C2S3×Dic5S3×C5⋊2C8
kernelDic3×C5⋊2C8C6×C5⋊2C8Dic3×C20C2×C15⋊3C8C3×C5⋊2C8C15⋊3C8C10×Dic3C5×Dic3C2×C5⋊2C8C4×Dic3C5⋊2C8C2×C20C2×Dic3C2×C12C20C2×C10Dic3C12C10C2×C4C4C4C22C2
# reps11114441612214222168822228

Matrix representation of Dic3×C5⋊2C8 ►in GL5(𝔽241)

10000
0240000
0024000
0002401
0002400
,
10000
064000
006400
0007551
000126166
,
10000
018924000
01000
00010
00001
,
2110000
03822000
017220300
0002400
0000240

G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,240,240,0,0,0,1,0],[1,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,75,126,0,0,0,51,166],[1,0,0,0,0,0,189,1,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,1],[211,0,0,0,0,0,38,172,0,0,0,220,203,0,0,0,0,0,240,0,0,0,0,0,240] >;
 

Dic3×C5⋊2C8 in GAP, Magma, Sage, TeX 

{\rm Dic}_3\times C_5\rtimes_2C_8
 
% in TeX
 
G:=Group("Dic3xC5:2C8");
 
// GroupNames label
 
G:=SmallGroup(480,26);
 
// by ID
 
G=gap.SmallGroup(480,26);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,64,100,1356,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=c^5=d^8=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

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