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## G = Dic15.Q8order 480 = 25·3·5

### 1st non-split extension by Dic15 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic15.Q8
G = < a,b,c,d | a30=c4=1, b2=a15, d2=c2, bab-1=a-1, ac=ca, dad-1=a19, cbc-1=a15b, bd=db, dcd-1=a15c-1 >

Subgroups: 460 in 112 conjugacy classes, 46 normal (44 characteristic)
C1, C2, C3, C4, C22, C5, C6, C2×C4, C2×C4, C10, Dic3, C12, C2×C6, C15, C42, C4⋊C4, Dic5, C20, C2×C10, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C30, C42.C2, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C5×Dic3, C3×Dic5, Dic15, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C5×C4⋊C4, Dic3.Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, Dic5.Q8, Dic3×Dic5, Dic155C4, C6.Dic10, C3×C10.D4, C5×C4⋊Dic3, C30.4Q8, Dic15.Q8
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, C4○D4, D10, C22×S3, C42.C2, C22×D5, C4○D12, D42S3, S3×Q8, S3×D5, C4○D20, D42D5, Q8×D5, Dic3.Q8, C2×S3×D5, Dic5.Q8, D15⋊Q8, D125D5, Dic5.D6, Dic15.Q8

Smallest permutation representation of Dic15.Q8
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 31 16 46)(2 60 17 45)(3 59 18 44)(4 58 19 43)(5 57 20 42)(6 56 21 41)(7 55 22 40)(8 54 23 39)(9 53 24 38)(10 52 25 37)(11 51 26 36)(12 50 27 35)(13 49 28 34)(14 48 29 33)(15 47 30 32)(61 306 76 321)(62 305 77 320)(63 304 78 319)(64 303 79 318)(65 302 80 317)(66 301 81 316)(67 330 82 315)(68 329 83 314)(69 328 84 313)(70 327 85 312)(71 326 86 311)(72 325 87 310)(73 324 88 309)(74 323 89 308)(75 322 90 307)(91 235 106 220)(92 234 107 219)(93 233 108 218)(94 232 109 217)(95 231 110 216)(96 230 111 215)(97 229 112 214)(98 228 113 213)(99 227 114 212)(100 226 115 211)(101 225 116 240)(102 224 117 239)(103 223 118 238)(104 222 119 237)(105 221 120 236)(121 172 136 157)(122 171 137 156)(123 170 138 155)(124 169 139 154)(125 168 140 153)(126 167 141 152)(127 166 142 151)(128 165 143 180)(129 164 144 179)(130 163 145 178)(131 162 146 177)(132 161 147 176)(133 160 148 175)(134 159 149 174)(135 158 150 173)(181 249 196 264)(182 248 197 263)(183 247 198 262)(184 246 199 261)(185 245 200 260)(186 244 201 259)(187 243 202 258)(188 242 203 257)(189 241 204 256)(190 270 205 255)(191 269 206 254)(192 268 207 253)(193 267 208 252)(194 266 209 251)(195 265 210 250)(271 343 286 358)(272 342 287 357)(273 341 288 356)(274 340 289 355)(275 339 290 354)(276 338 291 353)(277 337 292 352)(278 336 293 351)(279 335 294 350)(280 334 295 349)(281 333 296 348)(282 332 297 347)(283 331 298 346)(284 360 299 345)(285 359 300 344)(361 400 376 415)(362 399 377 414)(363 398 378 413)(364 397 379 412)(365 396 380 411)(366 395 381 410)(367 394 382 409)(368 393 383 408)(369 392 384 407)(370 391 385 406)(371 420 386 405)(372 419 387 404)(373 418 388 403)(374 417 389 402)(375 416 390 401)(421 467 436 452)(422 466 437 451)(423 465 438 480)(424 464 439 479)(425 463 440 478)(426 462 441 477)(427 461 442 476)(428 460 443 475)(429 459 444 474)(430 458 445 473)(431 457 446 472)(432 456 447 471)(433 455 448 470)(434 454 449 469)(435 453 450 468)
(1 349 106 202)(2 350 107 203)(3 351 108 204)(4 352 109 205)(5 353 110 206)(6 354 111 207)(7 355 112 208)(8 356 113 209)(9 357 114 210)(10 358 115 181)(11 359 116 182)(12 360 117 183)(13 331 118 184)(14 332 119 185)(15 333 120 186)(16 334 91 187)(17 335 92 188)(18 336 93 189)(19 337 94 190)(20 338 95 191)(21 339 96 192)(22 340 97 193)(23 341 98 194)(24 342 99 195)(25 343 100 196)(26 344 101 197)(27 345 102 198)(28 346 103 199)(29 347 104 200)(30 348 105 201)(31 295 220 243)(32 296 221 244)(33 297 222 245)(34 298 223 246)(35 299 224 247)(36 300 225 248)(37 271 226 249)(38 272 227 250)(39 273 228 251)(40 274 229 252)(41 275 230 253)(42 276 231 254)(43 277 232 255)(44 278 233 256)(45 279 234 257)(46 280 235 258)(47 281 236 259)(48 282 237 260)(49 283 238 261)(50 284 239 262)(51 285 240 263)(52 286 211 264)(53 287 212 265)(54 288 213 266)(55 289 214 267)(56 290 215 268)(57 291 216 269)(58 292 217 270)(59 293 218 241)(60 294 219 242)(61 178 431 364)(62 179 432 365)(63 180 433 366)(64 151 434 367)(65 152 435 368)(66 153 436 369)(67 154 437 370)(68 155 438 371)(69 156 439 372)(70 157 440 373)(71 158 441 374)(72 159 442 375)(73 160 443 376)(74 161 444 377)(75 162 445 378)(76 163 446 379)(77 164 447 380)(78 165 448 381)(79 166 449 382)(80 167 450 383)(81 168 421 384)(82 169 422 385)(83 170 423 386)(84 171 424 387)(85 172 425 388)(86 173 426 389)(87 174 427 390)(88 175 428 361)(89 176 429 362)(90 177 430 363)(121 463 418 312)(122 464 419 313)(123 465 420 314)(124 466 391 315)(125 467 392 316)(126 468 393 317)(127 469 394 318)(128 470 395 319)(129 471 396 320)(130 472 397 321)(131 473 398 322)(132 474 399 323)(133 475 400 324)(134 476 401 325)(135 477 402 326)(136 478 403 327)(137 479 404 328)(138 480 405 329)(139 451 406 330)(140 452 407 301)(141 453 408 302)(142 454 409 303)(143 455 410 304)(144 456 411 305)(145 457 412 306)(146 458 413 307)(147 459 414 308)(148 460 415 309)(149 461 416 310)(150 462 417 311)
(1 465 106 314)(2 454 107 303)(3 473 108 322)(4 462 109 311)(5 451 110 330)(6 470 111 319)(7 459 112 308)(8 478 113 327)(9 467 114 316)(10 456 115 305)(11 475 116 324)(12 464 117 313)(13 453 118 302)(14 472 119 321)(15 461 120 310)(16 480 91 329)(17 469 92 318)(18 458 93 307)(19 477 94 326)(20 466 95 315)(21 455 96 304)(22 474 97 323)(23 463 98 312)(24 452 99 301)(25 471 100 320)(26 460 101 309)(27 479 102 328)(28 468 103 317)(29 457 104 306)(30 476 105 325)(31 438 220 68)(32 427 221 87)(33 446 222 76)(34 435 223 65)(35 424 224 84)(36 443 225 73)(37 432 226 62)(38 421 227 81)(39 440 228 70)(40 429 229 89)(41 448 230 78)(42 437 231 67)(43 426 232 86)(44 445 233 75)(45 434 234 64)(46 423 235 83)(47 442 236 72)(48 431 237 61)(49 450 238 80)(50 439 239 69)(51 428 240 88)(52 447 211 77)(53 436 212 66)(54 425 213 85)(55 444 214 74)(56 433 215 63)(57 422 216 82)(58 441 217 71)(59 430 218 90)(60 449 219 79)(121 209 418 356)(122 198 419 345)(123 187 420 334)(124 206 391 353)(125 195 392 342)(126 184 393 331)(127 203 394 350)(128 192 395 339)(129 181 396 358)(130 200 397 347)(131 189 398 336)(132 208 399 355)(133 197 400 344)(134 186 401 333)(135 205 402 352)(136 194 403 341)(137 183 404 360)(138 202 405 349)(139 191 406 338)(140 210 407 357)(141 199 408 346)(142 188 409 335)(143 207 410 354)(144 196 411 343)(145 185 412 332)(146 204 413 351)(147 193 414 340)(148 182 415 359)(149 201 416 348)(150 190 417 337)(151 242 367 294)(152 261 368 283)(153 250 369 272)(154 269 370 291)(155 258 371 280)(156 247 372 299)(157 266 373 288)(158 255 374 277)(159 244 375 296)(160 263 376 285)(161 252 377 274)(162 241 378 293)(163 260 379 282)(164 249 380 271)(165 268 381 290)(166 257 382 279)(167 246 383 298)(168 265 384 287)(169 254 385 276)(170 243 386 295)(171 262 387 284)(172 251 388 273)(173 270 389 292)(174 259 390 281)(175 248 361 300)(176 267 362 289)(177 256 363 278)(178 245 364 297)(179 264 365 286)(180 253 366 275)

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

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

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

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 4 6 6 10 10 12 20 30 30 60 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 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 S3 Q8 D5 D6 D6 C4○D4 D10 D10 C4○D12 C4○D20 D4⋊2S3 S3×Q8 S3×D5 D4⋊2D5 Q8×D5 C2×S3×D5 D15⋊Q8 D12⋊5D5 Dic5.D6 kernel Dic15.Q8 Dic3×Dic5 Dic15⋊5C4 C6.Dic10 C3×C10.D4 C5×C4⋊Dic3 C30.4Q8 C10.D4 Dic15 C4⋊Dic3 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 C10 C6 C10 C10 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 2 1 1 1 1 1 2 2 2 1 4 4 2 4 8 1 1 2 2 2 2 4 4 4

Matrix representation of Dic15.Q8 in GL6(𝔽61)

 1 60 0 0 0 0 1 0 0 0 0 0 0 0 17 1 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 10 26 0 0 0 0 36 51 0 0 0 0 0 0 41 34 0 0 0 0 8 20 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 38 46 0 0 0 0 15 23 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 8 3 0 0 0 0 19 53
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 20 27 0 0 0 0 53 41 0 0 0 0 0 0 27 33 0 0 0 0 13 34

G:=sub<GL(6,GF(61))| [1,1,0,0,0,0,60,0,0,0,0,0,0,0,17,60,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[10,36,0,0,0,0,26,51,0,0,0,0,0,0,41,8,0,0,0,0,34,20,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[38,15,0,0,0,0,46,23,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,8,19,0,0,0,0,3,53],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,20,53,0,0,0,0,27,41,0,0,0,0,0,0,27,13,0,0,0,0,33,34] >;

Dic15.Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}.Q_8
% in TeX

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

G:=SmallGroup(480,412);
// by ID

G=gap.SmallGroup(480,412);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,254,219,100,1356,18822]);
// 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^19,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=a^15*c^-1>;
// generators/relations

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