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## G = Dic15⋊5Q8order 480 = 25·3·5

### 3rd semidirect product of Dic15 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Dic15⋊5Q8
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic3×Dic5 — Dic15⋊5Q8
 Lower central C15 — C30 — Dic15⋊5Q8
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic155Q8
G = < a,b,c,d | a30=c4=1, b2=a15, d2=c2, bab-1=cac-1=a-1, dad-1=a11, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 556 in 140 conjugacy classes, 60 normal (44 characteristic)
C1, C2, C3, C4, C22, C5, C6, C2×C4, C2×C4, Q8, C10, Dic3, Dic3, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C4×Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C3×C4⋊C4, C2×Dic6, C5×Dic3, C5×Dic3, C3×Dic5, C3×Dic5, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C10.D4, C5×C4⋊C4, C2×Dic10, Dic6⋊C4, C15⋊Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, Dic53Q8, Dic3×Dic5, Dic155C4, C3×C10.D4, C5×Dic3⋊C4, C4×Dic15, C2×C15⋊Q8, Dic155Q8
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D5, D6, C22×C4, C2×Q8, C4○D4, D10, C4×S3, C22×S3, C4×Q8, C4×D5, C22×D5, S3×C2×C4, D42S3, S3×Q8, S3×D5, C2×C4×D5, D42D5, Q8×D5, Dic6⋊C4, C2×S3×D5, Dic53Q8, D15⋊Q8, C4×S3×D5, C30.C23, Dic155Q8

Smallest permutation representation of Dic155Q8
Regular action on 480 points
Generators in S480
(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)
(1 424 16 439)(2 423 17 438)(3 422 18 437)(4 421 19 436)(5 450 20 435)(6 449 21 434)(7 448 22 433)(8 447 23 432)(9 446 24 431)(10 445 25 430)(11 444 26 429)(12 443 27 428)(13 442 28 427)(14 441 29 426)(15 440 30 425)(31 282 46 297)(32 281 47 296)(33 280 48 295)(34 279 49 294)(35 278 50 293)(36 277 51 292)(37 276 52 291)(38 275 53 290)(39 274 54 289)(40 273 55 288)(41 272 56 287)(42 271 57 286)(43 300 58 285)(44 299 59 284)(45 298 60 283)(61 195 76 210)(62 194 77 209)(63 193 78 208)(64 192 79 207)(65 191 80 206)(66 190 81 205)(67 189 82 204)(68 188 83 203)(69 187 84 202)(70 186 85 201)(71 185 86 200)(72 184 87 199)(73 183 88 198)(74 182 89 197)(75 181 90 196)(91 456 106 471)(92 455 107 470)(93 454 108 469)(94 453 109 468)(95 452 110 467)(96 451 111 466)(97 480 112 465)(98 479 113 464)(99 478 114 463)(100 477 115 462)(101 476 116 461)(102 475 117 460)(103 474 118 459)(104 473 119 458)(105 472 120 457)(121 307 136 322)(122 306 137 321)(123 305 138 320)(124 304 139 319)(125 303 140 318)(126 302 141 317)(127 301 142 316)(128 330 143 315)(129 329 144 314)(130 328 145 313)(131 327 146 312)(132 326 147 311)(133 325 148 310)(134 324 149 309)(135 323 150 308)(151 402 166 417)(152 401 167 416)(153 400 168 415)(154 399 169 414)(155 398 170 413)(156 397 171 412)(157 396 172 411)(158 395 173 410)(159 394 174 409)(160 393 175 408)(161 392 176 407)(162 391 177 406)(163 420 178 405)(164 419 179 404)(165 418 180 403)(211 242 226 257)(212 241 227 256)(213 270 228 255)(214 269 229 254)(215 268 230 253)(216 267 231 252)(217 266 232 251)(218 265 233 250)(219 264 234 249)(220 263 235 248)(221 262 236 247)(222 261 237 246)(223 260 238 245)(224 259 239 244)(225 258 240 243)(331 364 346 379)(332 363 347 378)(333 362 348 377)(334 361 349 376)(335 390 350 375)(336 389 351 374)(337 388 352 373)(338 387 353 372)(339 386 354 371)(340 385 355 370)(341 384 356 369)(342 383 357 368)(343 382 358 367)(344 381 359 366)(345 380 360 365)
(1 42 189 335)(2 41 190 334)(3 40 191 333)(4 39 192 332)(5 38 193 331)(6 37 194 360)(7 36 195 359)(8 35 196 358)(9 34 197 357)(10 33 198 356)(11 32 199 355)(12 31 200 354)(13 60 201 353)(14 59 202 352)(15 58 203 351)(16 57 204 350)(17 56 205 349)(18 55 206 348)(19 54 207 347)(20 53 208 346)(21 52 209 345)(22 51 210 344)(23 50 181 343)(24 49 182 342)(25 48 183 341)(26 47 184 340)(27 46 185 339)(28 45 186 338)(29 44 187 337)(30 43 188 336)(61 381 433 292)(62 380 434 291)(63 379 435 290)(64 378 436 289)(65 377 437 288)(66 376 438 287)(67 375 439 286)(68 374 440 285)(69 373 441 284)(70 372 442 283)(71 371 443 282)(72 370 444 281)(73 369 445 280)(74 368 446 279)(75 367 447 278)(76 366 448 277)(77 365 449 276)(78 364 450 275)(79 363 421 274)(80 362 422 273)(81 361 423 272)(82 390 424 271)(83 389 425 300)(84 388 426 299)(85 387 427 298)(86 386 428 297)(87 385 429 296)(88 384 430 295)(89 383 431 294)(90 382 432 293)(91 121 153 245)(92 150 154 244)(93 149 155 243)(94 148 156 242)(95 147 157 241)(96 146 158 270)(97 145 159 269)(98 144 160 268)(99 143 161 267)(100 142 162 266)(101 141 163 265)(102 140 164 264)(103 139 165 263)(104 138 166 262)(105 137 167 261)(106 136 168 260)(107 135 169 259)(108 134 170 258)(109 133 171 257)(110 132 172 256)(111 131 173 255)(112 130 174 254)(113 129 175 253)(114 128 176 252)(115 127 177 251)(116 126 178 250)(117 125 179 249)(118 124 180 248)(119 123 151 247)(120 122 152 246)(211 468 325 412)(212 467 326 411)(213 466 327 410)(214 465 328 409)(215 464 329 408)(216 463 330 407)(217 462 301 406)(218 461 302 405)(219 460 303 404)(220 459 304 403)(221 458 305 402)(222 457 306 401)(223 456 307 400)(224 455 308 399)(225 454 309 398)(226 453 310 397)(227 452 311 396)(228 451 312 395)(229 480 313 394)(230 479 314 393)(231 478 315 392)(232 477 316 391)(233 476 317 420)(234 475 318 419)(235 474 319 418)(236 473 320 417)(237 472 321 416)(238 471 322 415)(239 470 323 414)(240 469 324 413)
(1 267 189 143)(2 248 190 124)(3 259 191 135)(4 270 192 146)(5 251 193 127)(6 262 194 138)(7 243 195 149)(8 254 196 130)(9 265 197 141)(10 246 198 122)(11 257 199 133)(12 268 200 144)(13 249 201 125)(14 260 202 136)(15 241 203 147)(16 252 204 128)(17 263 205 139)(18 244 206 150)(19 255 207 131)(20 266 208 142)(21 247 209 123)(22 258 210 134)(23 269 181 145)(24 250 182 126)(25 261 183 137)(26 242 184 148)(27 253 185 129)(28 264 186 140)(29 245 187 121)(30 256 188 132)(31 160 354 98)(32 171 355 109)(33 152 356 120)(34 163 357 101)(35 174 358 112)(36 155 359 93)(37 166 360 104)(38 177 331 115)(39 158 332 96)(40 169 333 107)(41 180 334 118)(42 161 335 99)(43 172 336 110)(44 153 337 91)(45 164 338 102)(46 175 339 113)(47 156 340 94)(48 167 341 105)(49 178 342 116)(50 159 343 97)(51 170 344 108)(52 151 345 119)(53 162 346 100)(54 173 347 111)(55 154 348 92)(56 165 349 103)(57 176 350 114)(58 157 351 95)(59 168 352 106)(60 179 353 117)(61 324 433 240)(62 305 434 221)(63 316 435 232)(64 327 436 213)(65 308 437 224)(66 319 438 235)(67 330 439 216)(68 311 440 227)(69 322 441 238)(70 303 442 219)(71 314 443 230)(72 325 444 211)(73 306 445 222)(74 317 446 233)(75 328 447 214)(76 309 448 225)(77 320 449 236)(78 301 450 217)(79 312 421 228)(80 323 422 239)(81 304 423 220)(82 315 424 231)(83 326 425 212)(84 307 426 223)(85 318 427 234)(86 329 428 215)(87 310 429 226)(88 321 430 237)(89 302 431 218)(90 313 432 229)(271 392 390 478)(272 403 361 459)(273 414 362 470)(274 395 363 451)(275 406 364 462)(276 417 365 473)(277 398 366 454)(278 409 367 465)(279 420 368 476)(280 401 369 457)(281 412 370 468)(282 393 371 479)(283 404 372 460)(284 415 373 471)(285 396 374 452)(286 407 375 463)(287 418 376 474)(288 399 377 455)(289 410 378 466)(290 391 379 477)(291 402 380 458)(292 413 381 469)(293 394 382 480)(294 405 383 461)(295 416 384 472)(296 397 385 453)(297 408 386 464)(298 419 387 475)(299 400 388 456)(300 411 389 467)

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

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

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

66 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 5A 5B 6A 6B 6C 10A ··· 10F 12A 12B 12C 12D 12E 12F 15A 15B 20A 20B 20C 20D 20E ··· 20L 30A ··· 30F 60A ··· 60H order 1 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 10 ··· 10 12 12 12 12 12 12 15 15 20 20 20 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 6 6 6 6 10 10 10 10 15 15 15 15 30 30 2 2 2 2 2 2 ··· 2 4 4 20 20 20 20 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 type + + + + + + + + - + + + + + - - + - - + - image C1 C2 C2 C2 C2 C2 C2 C4 S3 Q8 D5 D6 D6 C4○D4 D10 D10 C4×S3 C4×D5 D4⋊2S3 S3×Q8 S3×D5 D4⋊2D5 Q8×D5 C2×S3×D5 D15⋊Q8 C4×S3×D5 C30.C23 kernel Dic15⋊5Q8 Dic3×Dic5 Dic15⋊5C4 C3×C10.D4 C5×Dic3⋊C4 C4×Dic15 C2×C15⋊Q8 C15⋊Q8 C10.D4 Dic15 Dic3⋊C4 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 Dic5 Dic3 C10 C10 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 2 1 1 1 1 1 8 1 2 2 2 1 2 4 2 4 8 1 1 2 2 2 2 4 4 4

Matrix representation of Dic155Q8 in GL6(𝔽61)

 44 1 0 0 0 0 16 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 60 0 0 0 0 1 0
,
 47 56 0 0 0 0 39 14 0 0 0 0 0 0 50 0 0 0 0 0 0 50 0 0 0 0 0 0 50 11 0 0 0 0 0 11
,
 14 5 0 0 0 0 22 47 0 0 0 0 0 0 1 39 0 0 0 0 50 60 0 0 0 0 0 0 60 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 19 0 0 0 0 32 1 0 0 0 0 0 0 1 60 0 0 0 0 0 60

G:=sub<GL(6,GF(61))| [44,16,0,0,0,0,1,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,1,0,0,0,0,60,0],[47,39,0,0,0,0,56,14,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,11,11],[14,22,0,0,0,0,5,47,0,0,0,0,0,0,1,50,0,0,0,0,39,60,0,0,0,0,0,0,60,0,0,0,0,0,1,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,32,0,0,0,0,19,1,0,0,0,0,0,0,1,0,0,0,0,0,60,60] >;

Dic155Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_5Q_8
% in TeX

G:=Group("Dic15:5Q8");
// GroupNames label

G:=SmallGroup(480,401);
// by ID

G=gap.SmallGroup(480,401);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,254,135,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^30=c^4=1,b^2=a^15,d^2=c^2,b*a*b^-1=c*a*c^-1=a^-1,d*a*d^-1=a^11,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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