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## G = Dic15.2Q8order 480 = 25·3·5

### 2nd non-split extension by Dic15 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — Dic15.2Q8
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic3×Dic5 — Dic15.2Q8
 Lower central C15 — C2×C30 — Dic15.2Q8
 Upper central C1 — C22 — C2×C4

Generators and relations for Dic15.2Q8
G = < a,b,c,d | a30=c4=1, b2=a15, d2=a15c2, bab-1=a-1, ac=ca, dad-1=a19, cbc-1=a15b, bd=db, dcd-1=c-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, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C5×Dic3, C3×Dic5, Dic15, Dic15, C60, C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, C5×C4⋊C4, Dic3.Q8, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, Dic5.Q8, Dic3×Dic5, C30.Q8, Dic155C4, C3×C4⋊Dic5, C5×Dic3⋊C4, C30.4Q8, Dic15.2Q8
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, D205S3, D15⋊Q8, Dic3.D10, Dic15.2Q8

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

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

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

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

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 D20⋊5S3 D15⋊Q8 Dic3.D10 kernel Dic15.2Q8 Dic3×Dic5 C30.Q8 Dic15⋊5C4 C3×C4⋊Dic5 C5×Dic3⋊C4 C30.4Q8 C4⋊Dic5 Dic15 Dic3⋊C4 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 C10 C6 C10 C10 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 1 2 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 Dic15.2Q8 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 34 0 0 0 0 0 49 9 0 0 0 0 0 0 0 60 0 0 0 0 1 1
,
 21 30 0 0 0 0 30 40 0 0 0 0 0 0 32 26 0 0 0 0 24 29 0 0 0 0 0 0 0 50 0 0 0 0 50 0
,
 0 1 0 0 0 0 60 0 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 52 43 0 0 0 0 18 9
,
 13 36 0 0 0 0 36 48 0 0 0 0 0 0 29 35 0 0 0 0 37 32 0 0 0 0 0 0 50 0 0 0 0 0 0 50

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,34,49,0,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,60,1],[21,30,0,0,0,0,30,40,0,0,0,0,0,0,32,24,0,0,0,0,26,29,0,0,0,0,0,0,0,50,0,0,0,0,50,0],[0,60,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,52,18,0,0,0,0,43,9],[13,36,0,0,0,0,36,48,0,0,0,0,0,0,29,37,0,0,0,0,35,32,0,0,0,0,0,0,50,0,0,0,0,0,0,50] >;

Dic15.2Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}._2Q_8
% in TeX

G:=Group("Dic15.2Q8");
// GroupNames label

G:=SmallGroup(480,415);
// by ID

G=gap.SmallGroup(480,415);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,254,219,58,1356,18822]);
// Polycyclic

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

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