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## G = Dic15⋊8Q8order 480 = 25·3·5

### 6th semidirect product of Dic15 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — Dic15⋊8Q8
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic15⋊5C4 — Dic15⋊8Q8
 Lower central C15 — C2×C30 — Dic15⋊8Q8
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic158Q8
G = < a,b,c,d | a30=c4=1, b2=a15, d2=c2, bab-1=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a15b, dcd-1=c-1 >

Subgroups: 540 in 136 conjugacy classes, 56 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, C2×C4, Q8, C10, C10, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, C20, C20, C2×C10, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C30, C30, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C4×Dic3, Dic3⋊C4, C2×Dic6, C6×Q8, C5×Dic3, C3×Dic5, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C2×Dic10, Q8×C10, Dic3⋊Q8, C3×Dic10, C6×Dic5, C5×Dic6, C10×Dic3, C2×Dic15, C2×C60, Dic5⋊Q8, Dic155C4, C4×Dic15, C6×Dic10, C10×Dic6, Dic158Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, D10, C3⋊D4, C22×S3, C4⋊Q8, C5⋊D4, C22×D5, S3×Q8, C2×C3⋊D4, S3×D5, Q8×D5, C2×C5⋊D4, Dic3⋊Q8, C15⋊D4, C2×S3×D5, Dic5⋊Q8, D15⋊Q8, C2×C15⋊D4, Dic158Q8

Smallest permutation representation of Dic158Q8
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)
(1 328 16 313)(2 327 17 312)(3 326 18 311)(4 325 19 310)(5 324 20 309)(6 323 21 308)(7 322 22 307)(8 321 23 306)(9 320 24 305)(10 319 25 304)(11 318 26 303)(12 317 27 302)(13 316 28 301)(14 315 29 330)(15 314 30 329)(31 418 46 403)(32 417 47 402)(33 416 48 401)(34 415 49 400)(35 414 50 399)(36 413 51 398)(37 412 52 397)(38 411 53 396)(39 410 54 395)(40 409 55 394)(41 408 56 393)(42 407 57 392)(43 406 58 391)(44 405 59 420)(45 404 60 419)(61 227 76 212)(62 226 77 211)(63 225 78 240)(64 224 79 239)(65 223 80 238)(66 222 81 237)(67 221 82 236)(68 220 83 235)(69 219 84 234)(70 218 85 233)(71 217 86 232)(72 216 87 231)(73 215 88 230)(74 214 89 229)(75 213 90 228)(91 284 106 299)(92 283 107 298)(93 282 108 297)(94 281 109 296)(95 280 110 295)(96 279 111 294)(97 278 112 293)(98 277 113 292)(99 276 114 291)(100 275 115 290)(101 274 116 289)(102 273 117 288)(103 272 118 287)(104 271 119 286)(105 300 120 285)(121 353 136 338)(122 352 137 337)(123 351 138 336)(124 350 139 335)(125 349 140 334)(126 348 141 333)(127 347 142 332)(128 346 143 331)(129 345 144 360)(130 344 145 359)(131 343 146 358)(132 342 147 357)(133 341 148 356)(134 340 149 355)(135 339 150 354)(151 201 166 186)(152 200 167 185)(153 199 168 184)(154 198 169 183)(155 197 170 182)(156 196 171 181)(157 195 172 210)(158 194 173 209)(159 193 174 208)(160 192 175 207)(161 191 176 206)(162 190 177 205)(163 189 178 204)(164 188 179 203)(165 187 180 202)(241 461 256 476)(242 460 257 475)(243 459 258 474)(244 458 259 473)(245 457 260 472)(246 456 261 471)(247 455 262 470)(248 454 263 469)(249 453 264 468)(250 452 265 467)(251 451 266 466)(252 480 267 465)(253 479 268 464)(254 478 269 463)(255 477 270 462)(361 423 376 438)(362 422 377 437)(363 421 378 436)(364 450 379 435)(365 449 380 434)(366 448 381 433)(367 447 382 432)(368 446 383 431)(369 445 384 430)(370 444 385 429)(371 443 386 428)(372 442 387 427)(373 441 388 426)(374 440 389 425)(375 439 390 424)
(1 119 183 267)(2 120 184 268)(3 91 185 269)(4 92 186 270)(5 93 187 241)(6 94 188 242)(7 95 189 243)(8 96 190 244)(9 97 191 245)(10 98 192 246)(11 99 193 247)(12 100 194 248)(13 101 195 249)(14 102 196 250)(15 103 197 251)(16 104 198 252)(17 105 199 253)(18 106 200 254)(19 107 201 255)(20 108 202 256)(21 109 203 257)(22 110 204 258)(23 111 205 259)(24 112 206 260)(25 113 207 261)(26 114 208 262)(27 115 209 263)(28 116 210 264)(29 117 181 265)(30 118 182 266)(31 213 362 335)(32 214 363 336)(33 215 364 337)(34 216 365 338)(35 217 366 339)(36 218 367 340)(37 219 368 341)(38 220 369 342)(39 221 370 343)(40 222 371 344)(41 223 372 345)(42 224 373 346)(43 225 374 347)(44 226 375 348)(45 227 376 349)(46 228 377 350)(47 229 378 351)(48 230 379 352)(49 231 380 353)(50 232 381 354)(51 233 382 355)(52 234 383 356)(53 235 384 357)(54 236 385 358)(55 237 386 359)(56 238 387 360)(57 239 388 331)(58 240 389 332)(59 211 390 333)(60 212 361 334)(61 423 125 419)(62 424 126 420)(63 425 127 391)(64 426 128 392)(65 427 129 393)(66 428 130 394)(67 429 131 395)(68 430 132 396)(69 431 133 397)(70 432 134 398)(71 433 135 399)(72 434 136 400)(73 435 137 401)(74 436 138 402)(75 437 139 403)(76 438 140 404)(77 439 141 405)(78 440 142 406)(79 441 143 407)(80 442 144 408)(81 443 145 409)(82 444 146 410)(83 445 147 411)(84 446 148 412)(85 447 149 413)(86 448 150 414)(87 449 121 415)(88 450 122 416)(89 421 123 417)(90 422 124 418)(151 462 325 283)(152 463 326 284)(153 464 327 285)(154 465 328 286)(155 466 329 287)(156 467 330 288)(157 468 301 289)(158 469 302 290)(159 470 303 291)(160 471 304 292)(161 472 305 293)(162 473 306 294)(163 474 307 295)(164 475 308 296)(165 476 309 297)(166 477 310 298)(167 478 311 299)(168 479 312 300)(169 480 313 271)(170 451 314 272)(171 452 315 273)(172 453 316 274)(173 454 317 275)(174 455 318 276)(175 456 319 277)(176 457 320 278)(177 458 321 279)(178 459 322 280)(179 460 323 281)(180 461 324 282)
(1 60 183 361)(2 41 184 372)(3 52 185 383)(4 33 186 364)(5 44 187 375)(6 55 188 386)(7 36 189 367)(8 47 190 378)(9 58 191 389)(10 39 192 370)(11 50 193 381)(12 31 194 362)(13 42 195 373)(14 53 196 384)(15 34 197 365)(16 45 198 376)(17 56 199 387)(18 37 200 368)(19 48 201 379)(20 59 202 390)(21 40 203 371)(22 51 204 382)(23 32 205 363)(24 43 206 374)(25 54 207 385)(26 35 208 366)(27 46 209 377)(28 57 210 388)(29 38 181 369)(30 49 182 380)(61 271 125 480)(62 282 126 461)(63 293 127 472)(64 274 128 453)(65 285 129 464)(66 296 130 475)(67 277 131 456)(68 288 132 467)(69 299 133 478)(70 280 134 459)(71 291 135 470)(72 272 136 451)(73 283 137 462)(74 294 138 473)(75 275 139 454)(76 286 140 465)(77 297 141 476)(78 278 142 457)(79 289 143 468)(80 300 144 479)(81 281 145 460)(82 292 146 471)(83 273 147 452)(84 284 148 463)(85 295 149 474)(86 276 150 455)(87 287 121 466)(88 298 122 477)(89 279 123 458)(90 290 124 469)(91 356 269 234)(92 337 270 215)(93 348 241 226)(94 359 242 237)(95 340 243 218)(96 351 244 229)(97 332 245 240)(98 343 246 221)(99 354 247 232)(100 335 248 213)(101 346 249 224)(102 357 250 235)(103 338 251 216)(104 349 252 227)(105 360 253 238)(106 341 254 219)(107 352 255 230)(108 333 256 211)(109 344 257 222)(110 355 258 233)(111 336 259 214)(112 347 260 225)(113 358 261 236)(114 339 262 217)(115 350 263 228)(116 331 264 239)(117 342 265 220)(118 353 266 231)(119 334 267 212)(120 345 268 223)(151 435 325 401)(152 446 326 412)(153 427 327 393)(154 438 328 404)(155 449 329 415)(156 430 330 396)(157 441 301 407)(158 422 302 418)(159 433 303 399)(160 444 304 410)(161 425 305 391)(162 436 306 402)(163 447 307 413)(164 428 308 394)(165 439 309 405)(166 450 310 416)(167 431 311 397)(168 442 312 408)(169 423 313 419)(170 434 314 400)(171 445 315 411)(172 426 316 392)(173 437 317 403)(174 448 318 414)(175 429 319 395)(176 440 320 406)(177 421 321 417)(178 432 322 398)(179 443 323 409)(180 424 324 420)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 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 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 12 12 20 20 30 30 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

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + - + + + + + + - + - - + image C1 C2 C2 C2 C2 S3 Q8 D4 D5 D6 D6 D10 D10 C3⋊D4 C5⋊D4 S3×Q8 S3×D5 Q8×D5 C15⋊D4 C2×S3×D5 D15⋊Q8 kernel Dic15⋊8Q8 Dic15⋊5C4 C4×Dic15 C6×Dic10 C10×Dic6 C2×Dic10 Dic15 C60 C2×Dic6 C2×Dic5 C2×C20 C2×Dic3 C2×C12 C20 C12 C10 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 1 4 2 2 2 1 4 2 4 8 2 2 4 4 2 8

Matrix representation of Dic158Q8 in GL6(𝔽61)

 2 41 0 0 0 0 52 60 0 0 0 0 0 0 0 1 0 0 0 0 60 43 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 44 7 0 0 0 0 37 17 0 0 0 0 0 0 59 47 0 0 0 0 22 2 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 19 5 0 0 0 0 13 42
,
 7 47 0 0 0 0 47 54 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 50 4 0 0 0 0 0 11

G:=sub<GL(6,GF(61))| [2,52,0,0,0,0,41,60,0,0,0,0,0,0,0,60,0,0,0,0,1,43,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[44,37,0,0,0,0,7,17,0,0,0,0,0,0,59,22,0,0,0,0,47,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,19,13,0,0,0,0,5,42],[7,47,0,0,0,0,47,54,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,50,0,0,0,0,0,4,11] >;

Dic158Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_8Q_8
% in TeX

G:=Group("Dic15:8Q8");
// GroupNames label

G:=SmallGroup(480,461);
// by ID

G=gap.SmallGroup(480,461);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,120,422,219,100,1356,18822]);
// Polycyclic

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

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