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## G = Dic15⋊4Q8order 480 = 25·3·5

### 2nd semidirect product of Dic15 and Q8 acting via Q8/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — Dic15⋊4Q8
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C2×Dic15 — C4×Dic15 — Dic15⋊4Q8
 Lower central C15 — C2×C30 — Dic15⋊4Q8
 Upper central C1 — C22 — C2×Q8

Generators and relations for Dic154Q8
G = < a,b,c,d | a30=c4=1, b2=a15, d2=c2, bab-1=a-1, ac=ca, ad=da, cbc-1=a15b, bd=db, dcd-1=c-1 >

Subgroups: 628 in 136 conjugacy classes, 59 normal (23 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, C2×C4, C2×C4, Q8, C10, C10, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, C2×Q8, Dic5, C20, C20, C2×C10, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C30, C30, C4⋊Q8, Dic10, C2×Dic5, C2×C20, C2×C20, C5×Q8, C4×Dic3, Dic3⋊C4, C2×Dic6, C6×Q8, Dic15, Dic15, C60, C60, C2×C30, C4×Dic5, C10.D4, C2×Dic10, Q8×C10, Dic3⋊Q8, Dic30, C2×Dic15, C2×C60, C2×C60, Q8×C15, Dic5⋊Q8, C4×Dic15, C30.4Q8, C2×Dic30, Q8×C30, Dic154Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, D10, C3⋊D4, C22×S3, D15, C4⋊Q8, C5⋊D4, C22×D5, S3×Q8, C2×C3⋊D4, D30, Q8×D5, C2×C5⋊D4, Dic3⋊Q8, C157D4, C22×D15, Dic5⋊Q8, Q8×D15, C2×C157D4, Dic154Q8

Smallest permutation representation of Dic154Q8
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)
(1 253 16 268)(2 252 17 267)(3 251 18 266)(4 250 19 265)(5 249 20 264)(6 248 21 263)(7 247 22 262)(8 246 23 261)(9 245 24 260)(10 244 25 259)(11 243 26 258)(12 242 27 257)(13 241 28 256)(14 270 29 255)(15 269 30 254)(31 165 46 180)(32 164 47 179)(33 163 48 178)(34 162 49 177)(35 161 50 176)(36 160 51 175)(37 159 52 174)(38 158 53 173)(39 157 54 172)(40 156 55 171)(41 155 56 170)(42 154 57 169)(43 153 58 168)(44 152 59 167)(45 151 60 166)(61 361 76 376)(62 390 77 375)(63 389 78 374)(64 388 79 373)(65 387 80 372)(66 386 81 371)(67 385 82 370)(68 384 83 369)(69 383 84 368)(70 382 85 367)(71 381 86 366)(72 380 87 365)(73 379 88 364)(74 378 89 363)(75 377 90 362)(91 290 106 275)(92 289 107 274)(93 288 108 273)(94 287 109 272)(95 286 110 271)(96 285 111 300)(97 284 112 299)(98 283 113 298)(99 282 114 297)(100 281 115 296)(101 280 116 295)(102 279 117 294)(103 278 118 293)(104 277 119 292)(105 276 120 291)(121 437 136 422)(122 436 137 421)(123 435 138 450)(124 434 139 449)(125 433 140 448)(126 432 141 447)(127 431 142 446)(128 430 143 445)(129 429 144 444)(130 428 145 443)(131 427 146 442)(132 426 147 441)(133 425 148 440)(134 424 149 439)(135 423 150 438)(181 336 196 351)(182 335 197 350)(183 334 198 349)(184 333 199 348)(185 332 200 347)(186 331 201 346)(187 360 202 345)(188 359 203 344)(189 358 204 343)(190 357 205 342)(191 356 206 341)(192 355 207 340)(193 354 208 339)(194 353 209 338)(195 352 210 337)(211 306 226 321)(212 305 227 320)(213 304 228 319)(214 303 229 318)(215 302 230 317)(216 301 231 316)(217 330 232 315)(218 329 233 314)(219 328 234 313)(220 327 235 312)(221 326 236 311)(222 325 237 310)(223 324 238 309)(224 323 239 308)(225 322 240 307)(391 460 406 475)(392 459 407 474)(393 458 408 473)(394 457 409 472)(395 456 410 471)(396 455 411 470)(397 454 412 469)(398 453 413 468)(399 452 414 467)(400 451 415 466)(401 480 416 465)(402 479 417 464)(403 478 418 463)(404 477 419 462)(405 476 420 461)
(1 356 283 422)(2 357 284 423)(3 358 285 424)(4 359 286 425)(5 360 287 426)(6 331 288 427)(7 332 289 428)(8 333 290 429)(9 334 291 430)(10 335 292 431)(11 336 293 432)(12 337 294 433)(13 338 295 434)(14 339 296 435)(15 340 297 436)(16 341 298 437)(17 342 299 438)(18 343 300 439)(19 344 271 440)(20 345 272 441)(21 346 273 442)(22 347 274 443)(23 348 275 444)(24 349 276 445)(25 350 277 446)(26 351 278 447)(27 352 279 448)(28 353 280 449)(29 354 281 450)(30 355 282 421)(31 324 380 391)(32 325 381 392)(33 326 382 393)(34 327 383 394)(35 328 384 395)(36 329 385 396)(37 330 386 397)(38 301 387 398)(39 302 388 399)(40 303 389 400)(41 304 390 401)(42 305 361 402)(43 306 362 403)(44 307 363 404)(45 308 364 405)(46 309 365 406)(47 310 366 407)(48 311 367 408)(49 312 368 409)(50 313 369 410)(51 314 370 411)(52 315 371 412)(53 316 372 413)(54 317 373 414)(55 318 374 415)(56 319 375 416)(57 320 376 417)(58 321 377 418)(59 322 378 419)(60 323 379 420)(61 479 169 227)(62 480 170 228)(63 451 171 229)(64 452 172 230)(65 453 173 231)(66 454 174 232)(67 455 175 233)(68 456 176 234)(69 457 177 235)(70 458 178 236)(71 459 179 237)(72 460 180 238)(73 461 151 239)(74 462 152 240)(75 463 153 211)(76 464 154 212)(77 465 155 213)(78 466 156 214)(79 467 157 215)(80 468 158 216)(81 469 159 217)(82 470 160 218)(83 471 161 219)(84 472 162 220)(85 473 163 221)(86 474 164 222)(87 475 165 223)(88 476 166 224)(89 477 167 225)(90 478 168 226)(91 144 261 199)(92 145 262 200)(93 146 263 201)(94 147 264 202)(95 148 265 203)(96 149 266 204)(97 150 267 205)(98 121 268 206)(99 122 269 207)(100 123 270 208)(101 124 241 209)(102 125 242 210)(103 126 243 181)(104 127 244 182)(105 128 245 183)(106 129 246 184)(107 130 247 185)(108 131 248 186)(109 132 249 187)(110 133 250 188)(111 134 251 189)(112 135 252 190)(113 136 253 191)(114 137 254 192)(115 138 255 193)(116 139 256 194)(117 140 257 195)(118 141 258 196)(119 142 259 197)(120 143 260 198)
(1 363 283 44)(2 364 284 45)(3 365 285 46)(4 366 286 47)(5 367 287 48)(6 368 288 49)(7 369 289 50)(8 370 290 51)(9 371 291 52)(10 372 292 53)(11 373 293 54)(12 374 294 55)(13 375 295 56)(14 376 296 57)(15 377 297 58)(16 378 298 59)(17 379 299 60)(18 380 300 31)(19 381 271 32)(20 382 272 33)(21 383 273 34)(22 384 274 35)(23 385 275 36)(24 386 276 37)(25 387 277 38)(26 388 278 39)(27 389 279 40)(28 390 280 41)(29 361 281 42)(30 362 282 43)(61 100 169 270)(62 101 170 241)(63 102 171 242)(64 103 172 243)(65 104 173 244)(66 105 174 245)(67 106 175 246)(68 107 176 247)(69 108 177 248)(70 109 178 249)(71 110 179 250)(72 111 180 251)(73 112 151 252)(74 113 152 253)(75 114 153 254)(76 115 154 255)(77 116 155 256)(78 117 156 257)(79 118 157 258)(80 119 158 259)(81 120 159 260)(82 91 160 261)(83 92 161 262)(84 93 162 263)(85 94 163 264)(86 95 164 265)(87 96 165 266)(88 97 166 267)(89 98 167 268)(90 99 168 269)(121 477 206 225)(122 478 207 226)(123 479 208 227)(124 480 209 228)(125 451 210 229)(126 452 181 230)(127 453 182 231)(128 454 183 232)(129 455 184 233)(130 456 185 234)(131 457 186 235)(132 458 187 236)(133 459 188 237)(134 460 189 238)(135 461 190 239)(136 462 191 240)(137 463 192 211)(138 464 193 212)(139 465 194 213)(140 466 195 214)(141 467 196 215)(142 468 197 216)(143 469 198 217)(144 470 199 218)(145 471 200 219)(146 472 201 220)(147 473 202 221)(148 474 203 222)(149 475 204 223)(150 476 205 224)(301 446 398 350)(302 447 399 351)(303 448 400 352)(304 449 401 353)(305 450 402 354)(306 421 403 355)(307 422 404 356)(308 423 405 357)(309 424 406 358)(310 425 407 359)(311 426 408 360)(312 427 409 331)(313 428 410 332)(314 429 411 333)(315 430 412 334)(316 431 413 335)(317 432 414 336)(318 433 415 337)(319 434 416 338)(320 435 417 339)(321 436 418 340)(322 437 419 341)(323 438 420 342)(324 439 391 343)(325 440 392 344)(326 441 393 345)(327 442 394 346)(328 443 395 347)(329 444 396 348)(330 445 397 349)

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

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

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

84 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 ··· 12F 15A 15B 15C 15D 20A ··· 20L 30A ··· 30L 60A ··· 60X 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 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 2 2 2 4 4 30 30 30 30 60 60 2 2 2 2 2 2 ··· 2 4 ··· 4 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

84 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + - + + + + + + - - - image C1 C2 C2 C2 C2 S3 Q8 D4 D5 D6 D10 C3⋊D4 D15 C5⋊D4 D30 C15⋊7D4 S3×Q8 Q8×D5 Q8×D15 kernel Dic15⋊4Q8 C4×Dic15 C30.4Q8 C2×Dic30 Q8×C30 Q8×C10 Dic15 C60 C6×Q8 C2×C20 C2×C12 C20 C2×Q8 C12 C2×C4 C4 C10 C6 C2 # reps 1 1 4 1 1 1 4 2 2 3 6 4 4 8 12 16 2 4 8

Matrix representation of Dic154Q8 in GL6(𝔽61)

 33 24 0 0 0 0 37 14 0 0 0 0 0 0 58 15 0 0 0 0 52 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 53 55 0 0 0 0 41 8 0 0 0 0 0 0 28 3 0 0 0 0 23 33 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 35 0 0 0 0 40 59 0 0 0 0 0 0 1 53 0 0 0 0 46 60
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 30 7 0 0 0 0 2 31

G:=sub<GL(6,GF(61))| [33,37,0,0,0,0,24,14,0,0,0,0,0,0,58,52,0,0,0,0,15,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[53,41,0,0,0,0,55,8,0,0,0,0,0,0,28,23,0,0,0,0,3,33,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,40,0,0,0,0,35,59,0,0,0,0,0,0,1,46,0,0,0,0,53,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,30,2,0,0,0,0,7,31] >;

Dic154Q8 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_4Q_8
% in TeX

G:=Group("Dic15:4Q8");
// GroupNames label

G:=SmallGroup(480,909);
// by ID

G=gap.SmallGroup(480,909);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,422,135,58,2693,18822]);
// 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,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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